KILLED
proof of input_nbe1p8WQpN.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 399 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 441 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
    (19) CompletionProof [UPPER BOUND(ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
        (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
(23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(24) CdtProblem
    (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (36) CdtProblem
    (37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 30 ms]
    (46) CdtProblem
    (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 35 ms]
    (50) CdtProblem
    (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (60) CdtProblem
    (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 52 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 13 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 50 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 86 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 11 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 134 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 26 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 94 ms]
    (98) CdtProblem
    (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 163 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 19 ms]
    (106) CdtProblem
    (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 114 ms]
    (110) CdtProblem
    (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 19 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 217 ms]
    (116) CdtProblem
    (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 3 ms]
    (118) CdtProblem
    (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 39 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 125 ms]
    (122) CdtProblem
    (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 13 ms]
    (124) CdtProblem
    (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 48 ms]
    (128) CdtProblem
    (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 5 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 62 ms]
    (134) CdtProblem
    (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 48 ms]
    (136) CdtProblem
    (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 38 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 96 ms]
    (140) CdtProblem
    (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 47 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 179 ms]
    (146) CdtProblem
    (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 190 ms]
    (150) CdtProblem
    (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms]
    (152) CdtProblem
    (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 244 ms]
    (154) CdtProblem
    (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 39 ms]
    (156) CdtProblem
    (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 331 ms]
    (158) CdtProblem
    (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 48 ms]
    (160) CdtProblem
    (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 262 ms]
    (162) CdtProblem
    (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 53 ms]
    (164) CdtProblem
    (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 462 ms]
    (166) CdtProblem
    (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 46 ms]
    (168) CdtProblem
    (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 292 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 286 ms]
    (172) CdtProblem
    (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 79 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 332 ms]
    (176) CdtProblem
    (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 79 ms]
    (178) CdtProblem
    (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 395 ms]
    (180) CdtProblem
    (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 86 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 505 ms]
    (184) CdtProblem
    (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 71 ms]
    (186) CdtProblem
    (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 356 ms]
    (188) CdtProblem
    (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 109 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 553 ms]
    (192) CdtProblem
    (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 91 ms]
    (194) CdtProblem
    (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 359 ms]
    (196) CdtProblem
    (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 139 ms]
    (198) CdtProblem
    (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 458 ms]
    (200) CdtProblem
    (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 136 ms]
    (202) CdtProblem
    (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 533 ms]
    (204) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X))))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
   a__prod(0, X) -> 0
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y)))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__zero(0) -> true
   a__zero(s(X)) -> false
   a__p(s(X)) -> mark(X)
   mark(fact(X)) -> a__fact(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(zero(X)) -> a__zero(mark(X))
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2))
   mark(p(X)) -> a__p(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
   a__fact(X) -> fact(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__zero(X) -> zero(X)
   a__prod(X1, X2) -> prod(X1, X2)
   a__p(X) -> p(X)
   a__add(X1, X2) -> add(X1, X2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0'), prod(X, fact(p(X))))
   a__add(0', X) -> mark(X)
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
   a__prod(0', X) -> 0'
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y)))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__zero(0') -> true
   a__zero(s(X)) -> false
   a__p(s(X)) -> mark(X)
   mark(fact(X)) -> a__fact(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(zero(X)) -> a__zero(mark(X))
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2))
   mark(p(X)) -> a__p(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(s(X)) -> s(mark(X))
   mark(0') -> 0'
   mark(true) -> true
   mark(false) -> false
   a__fact(X) -> fact(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__zero(X) -> zero(X)
   a__prod(X1, X2) -> prod(X1, X2)
   a__p(X) -> p(X)
   a__add(X1, X2) -> add(X1, X2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X))))
   a__add(0, X) -> mark(X)
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
   a__prod(0, X) -> 0
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y)))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   a__zero(0) -> true
   a__zero(s(X)) -> false
   a__p(s(X)) -> mark(X)
   mark(fact(X)) -> a__fact(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3)
   mark(zero(X)) -> a__zero(mark(X))
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2))
   mark(p(X)) -> a__p(mark(X))
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
   mark(s(X)) -> s(mark(X))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
   a__fact(X) -> fact(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)
   a__zero(X) -> zero(X)
   a__prod(X1, X2) -> prod(X1, X2)
   a__p(X) -> p(X)
   a__add(X1, X2) -> add(X1, X2)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) [1]
   a__add(0, X) -> mark(X) [1]
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) [1]
   a__prod(0, X) -> 0 [1]
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) [1]
   a__if(true, X, Y) -> mark(X) [1]
   a__if(false, X, Y) -> mark(Y) [1]
   a__zero(0) -> true [1]
   a__zero(s(X)) -> false [1]
   a__p(s(X)) -> mark(X) [1]
   mark(fact(X)) -> a__fact(mark(X)) [1]
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1]
   mark(zero(X)) -> a__zero(mark(X)) [1]
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) [1]
   mark(p(X)) -> a__p(mark(X)) [1]
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) [1]
   mark(s(X)) -> s(mark(X)) [1]
   mark(0) -> 0 [1]
   mark(true) -> true [1]
   mark(false) -> false [1]
   a__fact(X) -> fact(X) [1]
   a__if(X1, X2, X3) -> if(X1, X2, X3) [1]
   a__zero(X) -> zero(X) [1]
   a__prod(X1, X2) -> prod(X1, X2) [1]
   a__p(X) -> p(X) [1]
   a__add(X1, X2) -> add(X1, X2) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) [1]
   a__add(0, X) -> mark(X) [1]
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) [1]
   a__prod(0, X) -> 0 [1]
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) [1]
   a__if(true, X, Y) -> mark(X) [1]
   a__if(false, X, Y) -> mark(Y) [1]
   a__zero(0) -> true [1]
   a__zero(s(X)) -> false [1]
   a__p(s(X)) -> mark(X) [1]
   mark(fact(X)) -> a__fact(mark(X)) [1]
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1]
   mark(zero(X)) -> a__zero(mark(X)) [1]
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) [1]
   mark(p(X)) -> a__p(mark(X)) [1]
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) [1]
   mark(s(X)) -> s(mark(X)) [1]
   mark(0) -> 0 [1]
   mark(true) -> true [1]
   mark(false) -> false [1]
   a__fact(X) -> fact(X) [1]
   a__if(X1, X2, X3) -> if(X1, X2, X3) [1]
   a__zero(X) -> zero(X) [1]
   a__prod(X1, X2) -> prod(X1, X2) [1]
   a__p(X) -> p(X) [1]
   a__add(X1, X2) -> add(X1, X2) [1]

The TRS has the following type information:
a__fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
mark :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
s :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
0 :: 0:s:p:fact:prod:true:false:if:zero:add
prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
true :: 0:s:p:fact:prod:true:false:if:zero:add
false :: 0:s:p:fact:prod:true:false:if:zero:add
a__p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   a__zero_1
   mark_1
   a__prod_2
   a__fact_1
   a__if_3
   a__p_1
   a__add_2

Due to the following rules being added:
none

And the following fresh constants: none

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) [1]
   a__add(0, X) -> mark(X) [1]
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) [1]
   a__prod(0, X) -> 0 [1]
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) [1]
   a__if(true, X, Y) -> mark(X) [1]
   a__if(false, X, Y) -> mark(Y) [1]
   a__zero(0) -> true [1]
   a__zero(s(X)) -> false [1]
   a__p(s(X)) -> mark(X) [1]
   mark(fact(X)) -> a__fact(mark(X)) [1]
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1]
   mark(zero(X)) -> a__zero(mark(X)) [1]
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) [1]
   mark(p(X)) -> a__p(mark(X)) [1]
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) [1]
   mark(s(X)) -> s(mark(X)) [1]
   mark(0) -> 0 [1]
   mark(true) -> true [1]
   mark(false) -> false [1]
   a__fact(X) -> fact(X) [1]
   a__if(X1, X2, X3) -> if(X1, X2, X3) [1]
   a__zero(X) -> zero(X) [1]
   a__prod(X1, X2) -> prod(X1, X2) [1]
   a__p(X) -> p(X) [1]
   a__add(X1, X2) -> add(X1, X2) [1]

The TRS has the following type information:
a__fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
mark :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
s :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
0 :: 0:s:p:fact:prod:true:false:if:zero:add
prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
true :: 0:s:p:fact:prod:true:false:if:zero:add
false :: 0:s:p:fact:prod:true:false:if:zero:add
a__p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__fact(fact(X')) -> a__if(a__zero(a__fact(mark(X'))), s(0), prod(fact(X'), fact(p(fact(X'))))) [2]
   a__fact(if(X1', X2', X3')) -> a__if(a__zero(a__if(mark(X1'), X2', X3')), s(0), prod(if(X1', X2', X3'), fact(p(if(X1', X2', X3'))))) [2]
   a__fact(zero(X'')) -> a__if(a__zero(a__zero(mark(X''))), s(0), prod(zero(X''), fact(p(zero(X''))))) [2]
   a__fact(prod(X1'', X2'')) -> a__if(a__zero(a__prod(mark(X1''), mark(X2''))), s(0), prod(prod(X1'', X2''), fact(p(prod(X1'', X2''))))) [2]
   a__fact(p(X4)) -> a__if(a__zero(a__p(mark(X4))), s(0), prod(p(X4), fact(p(p(X4))))) [2]
   a__fact(add(X11, X21)) -> a__if(a__zero(a__add(mark(X11), mark(X21))), s(0), prod(add(X11, X21), fact(p(add(X11, X21))))) [2]
   a__fact(s(X5)) -> a__if(a__zero(s(mark(X5))), s(0), prod(s(X5), fact(p(s(X5))))) [2]
   a__fact(0) -> a__if(a__zero(0), s(0), prod(0, fact(p(0)))) [2]
   a__fact(true) -> a__if(a__zero(true), s(0), prod(true, fact(p(true)))) [2]
   a__fact(false) -> a__if(a__zero(false), s(0), prod(false, fact(p(false)))) [2]
   a__add(0, X) -> mark(X) [1]
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) [1]
   a__prod(0, X) -> 0 [1]
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) [1]
   a__if(true, X, Y) -> mark(X) [1]
   a__if(false, X, Y) -> mark(Y) [1]
   a__zero(0) -> true [1]
   a__zero(s(X)) -> false [1]
   a__p(s(X)) -> mark(X) [1]
   mark(fact(fact(X172))) -> a__fact(a__fact(mark(X172))) [2]
   mark(fact(if(X173, X269, X328))) -> a__fact(a__if(mark(X173), X269, X328)) [2]
   mark(fact(zero(X174))) -> a__fact(a__zero(mark(X174))) [2]
   mark(fact(prod(X175, X270))) -> a__fact(a__prod(mark(X175), mark(X270))) [2]
   mark(fact(p(X176))) -> a__fact(a__p(mark(X176))) [2]
   mark(fact(add(X177, X271))) -> a__fact(a__add(mark(X177), mark(X271))) [2]
   mark(fact(s(X178))) -> a__fact(s(mark(X178))) [2]
   mark(fact(0)) -> a__fact(0) [2]
   mark(fact(true)) -> a__fact(true) [2]
   mark(fact(false)) -> a__fact(false) [2]
   mark(if(fact(X179), X2, X3)) -> a__if(a__fact(mark(X179)), X2, X3) [2]
   mark(if(if(X180, X272, X329), X2, X3)) -> a__if(a__if(mark(X180), X272, X329), X2, X3) [2]
   mark(if(zero(X181), X2, X3)) -> a__if(a__zero(mark(X181)), X2, X3) [2]
   mark(if(prod(X182, X273), X2, X3)) -> a__if(a__prod(mark(X182), mark(X273)), X2, X3) [2]
   mark(if(p(X183), X2, X3)) -> a__if(a__p(mark(X183)), X2, X3) [2]
   mark(if(add(X184, X274), X2, X3)) -> a__if(a__add(mark(X184), mark(X274)), X2, X3) [2]
   mark(if(s(X185), X2, X3)) -> a__if(s(mark(X185)), X2, X3) [2]
   mark(if(0, X2, X3)) -> a__if(0, X2, X3) [2]
   mark(if(true, X2, X3)) -> a__if(true, X2, X3) [2]
   mark(if(false, X2, X3)) -> a__if(false, X2, X3) [2]
   mark(zero(fact(X186))) -> a__zero(a__fact(mark(X186))) [2]
   mark(zero(if(X187, X275, X330))) -> a__zero(a__if(mark(X187), X275, X330)) [2]
   mark(zero(zero(X188))) -> a__zero(a__zero(mark(X188))) [2]
   mark(zero(prod(X189, X276))) -> a__zero(a__prod(mark(X189), mark(X276))) [2]
   mark(zero(p(X190))) -> a__zero(a__p(mark(X190))) [2]
   mark(zero(add(X191, X277))) -> a__zero(a__add(mark(X191), mark(X277))) [2]
   mark(zero(s(X192))) -> a__zero(s(mark(X192))) [2]
   mark(zero(0)) -> a__zero(0) [2]
   mark(zero(true)) -> a__zero(true) [2]
   mark(zero(false)) -> a__zero(false) [2]
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) [1]
   mark(p(fact(X355))) -> a__p(a__fact(mark(X355))) [2]
   mark(p(if(X1130, X2118, X356))) -> a__p(a__if(mark(X1130), X2118, X356)) [2]
   mark(p(zero(X357))) -> a__p(a__zero(mark(X357))) [2]
   mark(p(prod(X1131, X2119))) -> a__p(a__prod(mark(X1131), mark(X2119))) [2]
   mark(p(p(X358))) -> a__p(a__p(mark(X358))) [2]
   mark(p(add(X1132, X2120))) -> a__p(a__add(mark(X1132), mark(X2120))) [2]
   mark(p(s(X359))) -> a__p(s(mark(X359))) [2]
   mark(p(0)) -> a__p(0) [2]
   mark(p(true)) -> a__p(true) [2]
   mark(p(false)) -> a__p(false) [2]
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) [1]
   mark(s(X)) -> s(mark(X)) [1]
   mark(0) -> 0 [1]
   mark(true) -> true [1]
   mark(false) -> false [1]
   a__fact(X) -> fact(X) [1]
   a__if(X1, X2, X3) -> if(X1, X2, X3) [1]
   a__zero(X) -> zero(X) [1]
   a__prod(X1, X2) -> prod(X1, X2) [1]
   a__p(X) -> p(X) [1]
   a__add(X1, X2) -> add(X1, X2) [1]

The TRS has the following type information:
a__fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
mark :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
s :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
0 :: 0:s:p:fact:prod:true:false:if:zero:add
prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
true :: 0:s:p:fact:prod:true:false:if:zero:add
false :: 0:s:p:fact:prod:true:false:if:zero:add
a__p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   a__add(z, z') -{ 1 }-> mark(X) :|: z' = X, X >= 0, z = 0
   a__add(z, z') -{ 1 }-> 1 + a__add(mark(X), mark(Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__add(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__fact(z) -{ 2 }-> a__if(a__zero(a__zero(mark(X''))), 1 + 0, 1 + (1 + X'') + (1 + (1 + (1 + X'')))) :|: z = 1 + X'', X'' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__prod(mark(X1''), mark(X2''))), 1 + 0, 1 + (1 + X1'' + X2'') + (1 + (1 + (1 + X1'' + X2'')))) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__p(mark(X4))), 1 + 0, 1 + (1 + X4) + (1 + (1 + (1 + X4)))) :|: z = 1 + X4, X4 >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__if(mark(X1'), X2', X3')), 1 + 0, 1 + (1 + X1' + X2' + X3') + (1 + (1 + (1 + X1' + X2' + X3')))) :|: X3' >= 0, X2' >= 0, z = 1 + X1' + X2' + X3', X1' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__fact(mark(X'))), 1 + 0, 1 + (1 + X') + (1 + (1 + (1 + X')))) :|: X' >= 0, z = 1 + X'
   a__fact(z) -{ 2 }-> a__if(a__zero(a__add(mark(X11), mark(X21))), 1 + 0, 1 + (1 + X11 + X21) + (1 + (1 + (1 + X11 + X21)))) :|: X11 >= 0, X21 >= 0, z = 1 + X11 + X21
   a__fact(z) -{ 2 }-> a__if(a__zero(2), 1 + 0, 1 + 2 + (1 + (1 + 2))) :|: z = 2
   a__fact(z) -{ 2 }-> a__if(a__zero(1), 1 + 0, 1 + 1 + (1 + (1 + 1))) :|: z = 1
   a__fact(z) -{ 2 }-> a__if(a__zero(0), 1 + 0, 1 + 0 + (1 + (1 + 0))) :|: z = 0
   a__fact(z) -{ 2 }-> a__if(a__zero(1 + mark(X5)), 1 + 0, 1 + (1 + X5) + (1 + (1 + (1 + X5)))) :|: X5 >= 0, z = 1 + X5
   a__fact(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__if(z, z', z'') -{ 1 }-> mark(X) :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   a__p(z) -{ 1 }-> mark(X) :|: z = 1 + X, X >= 0
   a__p(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__prod(z, z') -{ 1 }-> a__add(mark(Y), a__prod(mark(X), mark(Y))) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   a__prod(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__zero(z) -{ 1 }-> 2 :|: z = 0
   a__zero(z) -{ 1 }-> 1 :|: z = 1 + X, X >= 0
   a__zero(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   mark(z) -{ 2 }-> a__zero(a__zero(mark(X188))) :|: X188 >= 0, z = 1 + (1 + X188)
   mark(z) -{ 2 }-> a__zero(a__prod(mark(X189), mark(X276))) :|: X276 >= 0, z = 1 + (1 + X189 + X276), X189 >= 0
   mark(z) -{ 2 }-> a__zero(a__p(mark(X190))) :|: z = 1 + (1 + X190), X190 >= 0
   mark(z) -{ 2 }-> a__zero(a__if(mark(X187), X275, X330)) :|: X187 >= 0, X275 >= 0, z = 1 + (1 + X187 + X275 + X330), X330 >= 0
   mark(z) -{ 2 }-> a__zero(a__fact(mark(X186))) :|: z = 1 + (1 + X186), X186 >= 0
   mark(z) -{ 2 }-> a__zero(a__add(mark(X191), mark(X277))) :|: z = 1 + (1 + X191 + X277), X191 >= 0, X277 >= 0
   mark(z) -{ 2 }-> a__zero(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__zero(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__zero(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__zero(1 + mark(X192)) :|: X192 >= 0, z = 1 + (1 + X192)
   mark(z) -{ 1 }-> a__prod(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 2 }-> a__p(a__zero(mark(X357))) :|: z = 1 + (1 + X357), X357 >= 0
   mark(z) -{ 2 }-> a__p(a__prod(mark(X1131), mark(X2119))) :|: z = 1 + (1 + X1131 + X2119), X2119 >= 0, X1131 >= 0
   mark(z) -{ 2 }-> a__p(a__p(mark(X358))) :|: X358 >= 0, z = 1 + (1 + X358)
   mark(z) -{ 2 }-> a__p(a__if(mark(X1130), X2118, X356)) :|: X2118 >= 0, X1130 >= 0, z = 1 + (1 + X1130 + X2118 + X356), X356 >= 0
   mark(z) -{ 2 }-> a__p(a__fact(mark(X355))) :|: z = 1 + (1 + X355), X355 >= 0
   mark(z) -{ 2 }-> a__p(a__add(mark(X1132), mark(X2120))) :|: z = 1 + (1 + X1132 + X2120), X2120 >= 0, X1132 >= 0
   mark(z) -{ 2 }-> a__p(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__p(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__p(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__p(1 + mark(X359)) :|: z = 1 + (1 + X359), X359 >= 0
   mark(z) -{ 2 }-> a__if(a__zero(mark(X181)), X2, X3) :|: X181 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X181) + X2 + X3
   mark(z) -{ 2 }-> a__if(a__prod(mark(X182), mark(X273)), X2, X3) :|: z = 1 + (1 + X182 + X273) + X2 + X3, X273 >= 0, X3 >= 0, X182 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__p(mark(X183)), X2, X3) :|: X183 >= 0, z = 1 + (1 + X183) + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__if(mark(X180), X272, X329), X2, X3) :|: z = 1 + (1 + X180 + X272 + X329) + X2 + X3, X272 >= 0, X329 >= 0, X3 >= 0, X2 >= 0, X180 >= 0
   mark(z) -{ 2 }-> a__if(a__fact(mark(X179)), X2, X3) :|: z = 1 + (1 + X179) + X2 + X3, X3 >= 0, X2 >= 0, X179 >= 0
   mark(z) -{ 2 }-> a__if(a__add(mark(X184), mark(X274)), X2, X3) :|: X184 >= 0, X274 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X184 + X274) + X2 + X3
   mark(z) -{ 2 }-> a__if(2, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1 + mark(X185), X2, X3) :|: X185 >= 0, X3 >= 0, z = 1 + (1 + X185) + X2 + X3, X2 >= 0
   mark(z) -{ 2 }-> a__fact(a__zero(mark(X174))) :|: X174 >= 0, z = 1 + (1 + X174)
   mark(z) -{ 2 }-> a__fact(a__prod(mark(X175), mark(X270))) :|: z = 1 + (1 + X175 + X270), X175 >= 0, X270 >= 0
   mark(z) -{ 2 }-> a__fact(a__p(mark(X176))) :|: X176 >= 0, z = 1 + (1 + X176)
   mark(z) -{ 2 }-> a__fact(a__if(mark(X173), X269, X328)) :|: X269 >= 0, X173 >= 0, X328 >= 0, z = 1 + (1 + X173 + X269 + X328)
   mark(z) -{ 2 }-> a__fact(a__fact(mark(X172))) :|: X172 >= 0, z = 1 + (1 + X172)
   mark(z) -{ 2 }-> a__fact(a__add(mark(X177), mark(X271))) :|: z = 1 + (1 + X177 + X271), X177 >= 0, X271 >= 0
   mark(z) -{ 2 }-> a__fact(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__fact(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__fact(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__fact(1 + mark(X178)) :|: z = 1 + (1 + X178), X178 >= 0
   mark(z) -{ 1 }-> a__add(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> 2 :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0


----------------------------------------

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   a__zero(z) -{ 1 }-> 1 :|: z = 1 + X, X >= 0
   a__zero(z) -{ 1 }-> 2 :|: z = 0
   a__zero(z) -{ 1 }-> 1 + X :|: X >= 0, z = X

----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   a__add(z, z') -{ 1 }-> mark(X) :|: z' = X, X >= 0, z = 0
   a__add(z, z') -{ 1 }-> 1 + a__add(mark(X), mark(Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__add(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__fact(z) -{ 2 }-> a__if(a__zero(a__zero(mark(X''))), 1 + 0, 1 + (1 + X'') + (1 + (1 + (1 + X'')))) :|: z = 1 + X'', X'' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__prod(mark(X1''), mark(X2''))), 1 + 0, 1 + (1 + X1'' + X2'') + (1 + (1 + (1 + X1'' + X2'')))) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__p(mark(X4))), 1 + 0, 1 + (1 + X4) + (1 + (1 + (1 + X4)))) :|: z = 1 + X4, X4 >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__if(mark(X1'), X2', X3')), 1 + 0, 1 + (1 + X1' + X2' + X3') + (1 + (1 + (1 + X1' + X2' + X3')))) :|: X3' >= 0, X2' >= 0, z = 1 + X1' + X2' + X3', X1' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__fact(mark(X'))), 1 + 0, 1 + (1 + X') + (1 + (1 + (1 + X')))) :|: X' >= 0, z = 1 + X'
   a__fact(z) -{ 2 }-> a__if(a__zero(a__add(mark(X11), mark(X21))), 1 + 0, 1 + (1 + X11 + X21) + (1 + (1 + (1 + X11 + X21)))) :|: X11 >= 0, X21 >= 0, z = 1 + X11 + X21
   a__fact(z) -{ 2 }-> a__if(a__zero(1 + mark(X5)), 1 + 0, 1 + (1 + X5) + (1 + (1 + (1 + X5)))) :|: X5 >= 0, z = 1 + X5
   a__fact(z) -{ 3 }-> a__if(2, 1 + 0, 1 + 0 + (1 + (1 + 0))) :|: z = 0, 0 = 0
   a__fact(z) -{ 3 }-> a__if(1, 1 + 0, 1 + 2 + (1 + (1 + 2))) :|: z = 2, 2 = 1 + X, X >= 0
   a__fact(z) -{ 3 }-> a__if(1, 1 + 0, 1 + 1 + (1 + (1 + 1))) :|: z = 1, 1 = 1 + X, X >= 0
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 2 + (1 + (1 + 2))) :|: z = 2, X >= 0, 2 = X
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 1 + (1 + (1 + 1))) :|: z = 1, X >= 0, 1 = X
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 0 + (1 + (1 + 0))) :|: z = 0, X >= 0, 0 = X
   a__fact(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__if(z, z', z'') -{ 1 }-> mark(X) :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   a__p(z) -{ 1 }-> mark(X) :|: z = 1 + X, X >= 0
   a__p(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__prod(z, z') -{ 1 }-> a__add(mark(Y), a__prod(mark(X), mark(Y))) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   a__prod(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__zero(z) -{ 1 }-> 2 :|: z = 0
   a__zero(z) -{ 1 }-> 1 :|: z = 1 + X, X >= 0
   a__zero(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   mark(z) -{ 2 }-> a__zero(a__zero(mark(X188))) :|: X188 >= 0, z = 1 + (1 + X188)
   mark(z) -{ 2 }-> a__zero(a__prod(mark(X189), mark(X276))) :|: X276 >= 0, z = 1 + (1 + X189 + X276), X189 >= 0
   mark(z) -{ 2 }-> a__zero(a__p(mark(X190))) :|: z = 1 + (1 + X190), X190 >= 0
   mark(z) -{ 2 }-> a__zero(a__if(mark(X187), X275, X330)) :|: X187 >= 0, X275 >= 0, z = 1 + (1 + X187 + X275 + X330), X330 >= 0
   mark(z) -{ 2 }-> a__zero(a__fact(mark(X186))) :|: z = 1 + (1 + X186), X186 >= 0
   mark(z) -{ 2 }-> a__zero(a__add(mark(X191), mark(X277))) :|: z = 1 + (1 + X191 + X277), X191 >= 0, X277 >= 0
   mark(z) -{ 2 }-> a__zero(1 + mark(X192)) :|: X192 >= 0, z = 1 + (1 + X192)
   mark(z) -{ 1 }-> a__prod(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 2 }-> a__p(a__zero(mark(X357))) :|: z = 1 + (1 + X357), X357 >= 0
   mark(z) -{ 2 }-> a__p(a__prod(mark(X1131), mark(X2119))) :|: z = 1 + (1 + X1131 + X2119), X2119 >= 0, X1131 >= 0
   mark(z) -{ 2 }-> a__p(a__p(mark(X358))) :|: X358 >= 0, z = 1 + (1 + X358)
   mark(z) -{ 2 }-> a__p(a__if(mark(X1130), X2118, X356)) :|: X2118 >= 0, X1130 >= 0, z = 1 + (1 + X1130 + X2118 + X356), X356 >= 0
   mark(z) -{ 2 }-> a__p(a__fact(mark(X355))) :|: z = 1 + (1 + X355), X355 >= 0
   mark(z) -{ 2 }-> a__p(a__add(mark(X1132), mark(X2120))) :|: z = 1 + (1 + X1132 + X2120), X2120 >= 0, X1132 >= 0
   mark(z) -{ 2 }-> a__p(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__p(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__p(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__p(1 + mark(X359)) :|: z = 1 + (1 + X359), X359 >= 0
   mark(z) -{ 2 }-> a__if(a__zero(mark(X181)), X2, X3) :|: X181 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X181) + X2 + X3
   mark(z) -{ 2 }-> a__if(a__prod(mark(X182), mark(X273)), X2, X3) :|: z = 1 + (1 + X182 + X273) + X2 + X3, X273 >= 0, X3 >= 0, X182 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__p(mark(X183)), X2, X3) :|: X183 >= 0, z = 1 + (1 + X183) + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__if(mark(X180), X272, X329), X2, X3) :|: z = 1 + (1 + X180 + X272 + X329) + X2 + X3, X272 >= 0, X329 >= 0, X3 >= 0, X2 >= 0, X180 >= 0
   mark(z) -{ 2 }-> a__if(a__fact(mark(X179)), X2, X3) :|: z = 1 + (1 + X179) + X2 + X3, X3 >= 0, X2 >= 0, X179 >= 0
   mark(z) -{ 2 }-> a__if(a__add(mark(X184), mark(X274)), X2, X3) :|: X184 >= 0, X274 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X184 + X274) + X2 + X3
   mark(z) -{ 2 }-> a__if(2, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1 + mark(X185), X2, X3) :|: X185 >= 0, X3 >= 0, z = 1 + (1 + X185) + X2 + X3, X2 >= 0
   mark(z) -{ 2 }-> a__fact(a__zero(mark(X174))) :|: X174 >= 0, z = 1 + (1 + X174)
   mark(z) -{ 2 }-> a__fact(a__prod(mark(X175), mark(X270))) :|: z = 1 + (1 + X175 + X270), X175 >= 0, X270 >= 0
   mark(z) -{ 2 }-> a__fact(a__p(mark(X176))) :|: X176 >= 0, z = 1 + (1 + X176)
   mark(z) -{ 2 }-> a__fact(a__if(mark(X173), X269, X328)) :|: X269 >= 0, X173 >= 0, X328 >= 0, z = 1 + (1 + X173 + X269 + X328)
   mark(z) -{ 2 }-> a__fact(a__fact(mark(X172))) :|: X172 >= 0, z = 1 + (1 + X172)
   mark(z) -{ 2 }-> a__fact(a__add(mark(X177), mark(X271))) :|: z = 1 + (1 + X177 + X271), X177 >= 0, X271 >= 0
   mark(z) -{ 2 }-> a__fact(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__fact(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__fact(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__fact(1 + mark(X178)) :|: z = 1 + (1 + X178), X178 >= 0
   mark(z) -{ 1 }-> a__add(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> 2 :|: z = 2
   mark(z) -{ 3 }-> 2 :|: z = 1 + 0, 0 = 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 3 }-> 1 :|: z = 1 + 2, 2 = 1 + X, X >= 0
   mark(z) -{ 3 }-> 1 :|: z = 1 + 1, 1 = 1 + X, X >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 0, X >= 0, 0 = X
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 2, X >= 0, 2 = X
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
   mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0


----------------------------------------

(17) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   a__add(z, z') -{ 1 }-> mark(z') :|: z' >= 0, z = 0
   a__add(z, z') -{ 1 }-> 1 + a__add(mark(z - 1), mark(z')) :|: z' >= 0, z - 1 >= 0
   a__add(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__zero(mark(z - 1))), 1 + 0, 1 + (1 + (z - 1)) + (1 + (1 + (1 + (z - 1))))) :|: z - 1 >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__prod(mark(X1''), mark(X2''))), 1 + 0, 1 + (1 + X1'' + X2'') + (1 + (1 + (1 + X1'' + X2'')))) :|: X1'' >= 0, z = 1 + X1'' + X2'', X2'' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__p(mark(z - 1))), 1 + 0, 1 + (1 + (z - 1)) + (1 + (1 + (1 + (z - 1))))) :|: z - 1 >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__if(mark(X1'), X2', X3')), 1 + 0, 1 + (1 + X1' + X2' + X3') + (1 + (1 + (1 + X1' + X2' + X3')))) :|: X3' >= 0, X2' >= 0, z = 1 + X1' + X2' + X3', X1' >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__fact(mark(z - 1))), 1 + 0, 1 + (1 + (z - 1)) + (1 + (1 + (1 + (z - 1))))) :|: z - 1 >= 0
   a__fact(z) -{ 2 }-> a__if(a__zero(a__add(mark(X11), mark(X21))), 1 + 0, 1 + (1 + X11 + X21) + (1 + (1 + (1 + X11 + X21)))) :|: X11 >= 0, X21 >= 0, z = 1 + X11 + X21
   a__fact(z) -{ 2 }-> a__if(a__zero(1 + mark(z - 1)), 1 + 0, 1 + (1 + (z - 1)) + (1 + (1 + (1 + (z - 1))))) :|: z - 1 >= 0
   a__fact(z) -{ 3 }-> a__if(2, 1 + 0, 1 + 0 + (1 + (1 + 0))) :|: z = 0, 0 = 0
   a__fact(z) -{ 3 }-> a__if(1, 1 + 0, 1 + 2 + (1 + (1 + 2))) :|: z = 2, 2 = 1 + X, X >= 0
   a__fact(z) -{ 3 }-> a__if(1, 1 + 0, 1 + 1 + (1 + (1 + 1))) :|: z = 1, 1 = 1 + X, X >= 0
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 2 + (1 + (1 + 2))) :|: z = 2, X >= 0, 2 = X
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 1 + (1 + (1 + 1))) :|: z = 1, X >= 0, 1 = X
   a__fact(z) -{ 3 }-> a__if(1 + X, 1 + 0, 1 + 0 + (1 + (1 + 0))) :|: z = 0, X >= 0, 0 = X
   a__fact(z) -{ 1 }-> 1 + z :|: z >= 0
   a__if(z, z', z'') -{ 1 }-> mark(z') :|: z = 2, z'' >= 0, z' >= 0
   a__if(z, z', z'') -{ 1 }-> mark(z'') :|: z'' >= 0, z = 1, z' >= 0
   a__if(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   a__p(z) -{ 1 }-> mark(z - 1) :|: z - 1 >= 0
   a__p(z) -{ 1 }-> 1 + z :|: z >= 0
   a__prod(z, z') -{ 1 }-> a__add(mark(z'), a__prod(mark(z - 1), mark(z'))) :|: z' >= 0, z - 1 >= 0
   a__prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a__prod(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   a__zero(z) -{ 1 }-> 2 :|: z = 0
   a__zero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   a__zero(z) -{ 1 }-> 1 + z :|: z >= 0
   mark(z) -{ 2 }-> a__zero(a__zero(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__zero(a__prod(mark(X189), mark(X276))) :|: X276 >= 0, z = 1 + (1 + X189 + X276), X189 >= 0
   mark(z) -{ 2 }-> a__zero(a__p(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__zero(a__if(mark(X187), X275, X330)) :|: X187 >= 0, X275 >= 0, z = 1 + (1 + X187 + X275 + X330), X330 >= 0
   mark(z) -{ 2 }-> a__zero(a__fact(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__zero(a__add(mark(X191), mark(X277))) :|: z = 1 + (1 + X191 + X277), X191 >= 0, X277 >= 0
   mark(z) -{ 2 }-> a__zero(1 + mark(z - 2)) :|: z - 2 >= 0
   mark(z) -{ 1 }-> a__prod(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 2 }-> a__p(a__zero(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__p(a__prod(mark(X1131), mark(X2119))) :|: z = 1 + (1 + X1131 + X2119), X2119 >= 0, X1131 >= 0
   mark(z) -{ 2 }-> a__p(a__p(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__p(a__if(mark(X1130), X2118, X356)) :|: X2118 >= 0, X1130 >= 0, z = 1 + (1 + X1130 + X2118 + X356), X356 >= 0
   mark(z) -{ 2 }-> a__p(a__fact(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__p(a__add(mark(X1132), mark(X2120))) :|: z = 1 + (1 + X1132 + X2120), X2120 >= 0, X1132 >= 0
   mark(z) -{ 2 }-> a__p(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__p(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__p(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__p(1 + mark(z - 2)) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__if(a__zero(mark(X181)), X2, X3) :|: X181 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X181) + X2 + X3
   mark(z) -{ 2 }-> a__if(a__prod(mark(X182), mark(X273)), X2, X3) :|: z = 1 + (1 + X182 + X273) + X2 + X3, X273 >= 0, X3 >= 0, X182 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__p(mark(X183)), X2, X3) :|: X183 >= 0, z = 1 + (1 + X183) + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(a__if(mark(X180), X272, X329), X2, X3) :|: z = 1 + (1 + X180 + X272 + X329) + X2 + X3, X272 >= 0, X329 >= 0, X3 >= 0, X2 >= 0, X180 >= 0
   mark(z) -{ 2 }-> a__if(a__fact(mark(X179)), X2, X3) :|: z = 1 + (1 + X179) + X2 + X3, X3 >= 0, X2 >= 0, X179 >= 0
   mark(z) -{ 2 }-> a__if(a__add(mark(X184), mark(X274)), X2, X3) :|: X184 >= 0, X274 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X184 + X274) + X2 + X3
   mark(z) -{ 2 }-> a__if(2, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0
   mark(z) -{ 2 }-> a__if(1 + mark(X185), X2, X3) :|: X185 >= 0, X3 >= 0, z = 1 + (1 + X185) + X2 + X3, X2 >= 0
   mark(z) -{ 2 }-> a__fact(a__zero(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__fact(a__prod(mark(X175), mark(X270))) :|: z = 1 + (1 + X175 + X270), X175 >= 0, X270 >= 0
   mark(z) -{ 2 }-> a__fact(a__p(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__fact(a__if(mark(X173), X269, X328)) :|: X269 >= 0, X173 >= 0, X328 >= 0, z = 1 + (1 + X173 + X269 + X328)
   mark(z) -{ 2 }-> a__fact(a__fact(mark(z - 2))) :|: z - 2 >= 0
   mark(z) -{ 2 }-> a__fact(a__add(mark(X177), mark(X271))) :|: z = 1 + (1 + X177 + X271), X177 >= 0, X271 >= 0
   mark(z) -{ 2 }-> a__fact(2) :|: z = 1 + 2
   mark(z) -{ 2 }-> a__fact(1) :|: z = 1 + 1
   mark(z) -{ 2 }-> a__fact(0) :|: z = 1 + 0
   mark(z) -{ 2 }-> a__fact(1 + mark(z - 2)) :|: z - 2 >= 0
   mark(z) -{ 1 }-> a__add(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> 2 :|: z = 2
   mark(z) -{ 3 }-> 2 :|: z = 1 + 0, 0 = 0
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 3 }-> 1 :|: z = 1 + 2, 2 = 1 + X, X >= 0
   mark(z) -{ 3 }-> 1 :|: z = 1 + 1, 1 = 1 + X, X >= 0
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 0, X >= 0, 0 = X
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 2, X >= 0, 2 = X
   mark(z) -{ 3 }-> 1 + X :|: z = 1 + 1, X >= 0, 1 = X
   mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0


----------------------------------------

(19) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

----------------------------------------

(20)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__fact(X) -> a__if(a__zero(mark(X)), s(0), prod(X, fact(p(X)))) [1]
   a__add(0, X) -> mark(X) [1]
   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) [1]
   a__prod(0, X) -> 0 [1]
   a__prod(s(X), Y) -> a__add(mark(Y), a__prod(mark(X), mark(Y))) [1]
   a__if(true, X, Y) -> mark(X) [1]
   a__if(false, X, Y) -> mark(Y) [1]
   a__zero(0) -> true [1]
   a__zero(s(X)) -> false [1]
   a__p(s(X)) -> mark(X) [1]
   mark(fact(X)) -> a__fact(mark(X)) [1]
   mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1]
   mark(zero(X)) -> a__zero(mark(X)) [1]
   mark(prod(X1, X2)) -> a__prod(mark(X1), mark(X2)) [1]
   mark(p(X)) -> a__p(mark(X)) [1]
   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) [1]
   mark(s(X)) -> s(mark(X)) [1]
   mark(0) -> 0 [1]
   mark(true) -> true [1]
   mark(false) -> false [1]
   a__fact(X) -> fact(X) [1]
   a__if(X1, X2, X3) -> if(X1, X2, X3) [1]
   a__zero(X) -> zero(X) [1]
   a__prod(X1, X2) -> prod(X1, X2) [1]
   a__p(X) -> p(X) [1]
   a__add(X1, X2) -> add(X1, X2) [1]

The TRS has the following type information:
a__fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
mark :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
s :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
0 :: 0:s:p:fact:prod:true:false:if:zero:add
prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
fact :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
a__prod :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
true :: 0:s:p:fact:prod:true:false:if:zero:add
false :: 0:s:p:fact:prod:true:false:if:zero:add
a__p :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
if :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
zero :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add
add :: 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add -> 0:s:p:fact:prod:true:false:if:zero:add

Rewrite Strategy: INNERMOST
----------------------------------------

(21) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   a__add(z, z') -{ 1 }-> mark(X) :|: z' = X, X >= 0, z = 0
   a__add(z, z') -{ 1 }-> 1 + a__add(mark(X), mark(Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__add(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__fact(z) -{ 1 }-> a__if(a__zero(mark(X)), 1 + 0, 1 + X + (1 + (1 + X))) :|: X >= 0, z = X
   a__fact(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__if(z, z', z'') -{ 1 }-> mark(X) :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   a__p(z) -{ 1 }-> mark(X) :|: z = 1 + X, X >= 0
   a__p(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   a__prod(z, z') -{ 1 }-> a__add(mark(Y), a__prod(mark(X), mark(Y))) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   a__prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0
   a__prod(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   a__zero(z) -{ 1 }-> 2 :|: z = 0
   a__zero(z) -{ 1 }-> 1 :|: z = 1 + X, X >= 0
   a__zero(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   mark(z) -{ 1 }-> a__zero(mark(X)) :|: z = 1 + X, X >= 0
   mark(z) -{ 1 }-> a__prod(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> a__p(mark(X)) :|: z = 1 + X, X >= 0
   mark(z) -{ 1 }-> a__if(mark(X1), X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> a__fact(mark(X)) :|: z = 1 + X, X >= 0
   mark(z) -{ 1 }-> a__add(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   mark(z) -{ 1 }-> 2 :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(23) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), A__ZERO(mark(z0)), MARK(z0))
   A__FACT(z0) -> c1
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__ADD(z0, z1) -> c5
   A__PROD(0, z0) -> c6
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(z0, z1) -> c10
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__IF(z0, z1, z2) -> c13
   A__ZERO(0) -> c14
   A__ZERO(s(z0)) -> c15
   A__ZERO(z0) -> c16
   A__P(s(z0)) -> c17(MARK(z0))
   A__P(z0) -> c18
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(zero(z0)) -> c21(A__ZERO(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(0) -> c28
   MARK(true) -> c29
   MARK(false) -> c30
S tuples:
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), A__ZERO(mark(z0)), MARK(z0))
   A__FACT(z0) -> c1
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__ADD(z0, z1) -> c5
   A__PROD(0, z0) -> c6
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(z0, z1) -> c10
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__IF(z0, z1, z2) -> c13
   A__ZERO(0) -> c14
   A__ZERO(s(z0)) -> c15
   A__ZERO(z0) -> c16
   A__P(s(z0)) -> c17(MARK(z0))
   A__P(z0) -> c18
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(zero(z0)) -> c21(A__ZERO(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(0) -> c28
   MARK(true) -> c29
   MARK(false) -> c30
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__FACT_1, A__ADD_2, A__PROD_2, A__IF_3, A__ZERO_1, A__P_1, MARK_1

Compound Symbols:   c_3, c1, c2_1, c3_2, c4_2, c5, c6, c7_2, c8_3, c9_3, c10, c11_1, c12_1, c13, c14, c15, c16, c17_1, c18, c19_2, c20_2, c21_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c28, c29, c30


----------------------------------------

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 12 trailing nodes:
   A__PROD(z0, z1) -> c10
   MARK(true) -> c29
   A__FACT(z0) -> c1
   MARK(0) -> c28
   A__ZERO(z0) -> c16
   MARK(false) -> c30
   A__PROD(0, z0) -> c6
   A__ZERO(s(z0)) -> c15
   A__ADD(z0, z1) -> c5
   A__P(z0) -> c18
   A__ZERO(0) -> c14
   A__IF(z0, z1, z2) -> c13

----------------------------------------

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), A__ZERO(mark(z0)), MARK(z0))
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(zero(z0)) -> c21(A__ZERO(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
S tuples:
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), A__ZERO(mark(z0)), MARK(z0))
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(zero(z0)) -> c21(A__ZERO(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__FACT_1, A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1

Compound Symbols:   c_3, c2_1, c3_2, c4_2, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c21_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1


----------------------------------------

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c3_2, c4_2, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1


----------------------------------------

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(z0), z1) -> c3(A__ADD(mark(z0), mark(z1)), MARK(z0)) by 
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)), MARK(0))
   A__ADD(s(true), x1) -> c3(A__ADD(true, mark(x1)), MARK(true))
   A__ADD(s(false), x1) -> c3(A__ADD(false, mark(x1)), MARK(false))

----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)), MARK(0))
   A__ADD(s(true), x1) -> c3(A__ADD(true, mark(x1)), MARK(true))
   A__ADD(s(false), x1) -> c3(A__ADD(false, mark(x1)), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)), MARK(0))
   A__ADD(s(true), x1) -> c3(A__ADD(true, mark(x1)), MARK(true))
   A__ADD(s(false), x1) -> c3(A__ADD(false, mark(x1)), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c4_2, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2


----------------------------------------

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), x1) -> c3(A__ADD(true, mark(x1)), MARK(true))
   A__ADD(s(false), x1) -> c3(A__ADD(false, mark(x1)), MARK(false))

----------------------------------------

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c4_2, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2


----------------------------------------

(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c4_2, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1


----------------------------------------

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(z0), z1) -> c4(A__ADD(mark(z0), mark(z1)), MARK(z1)) by 
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0), MARK(0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true), MARK(true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false), MARK(false))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(true), x1) -> c4(A__ADD(true, mark(x1)), MARK(x1))
   A__ADD(s(false), x1) -> c4(A__ADD(false, mark(x1)), MARK(x1))

----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0), MARK(0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true), MARK(true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false), MARK(false))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(true), x1) -> c4(A__ADD(true, mark(x1)), MARK(x1))
   A__ADD(s(false), x1) -> c4(A__ADD(false, mark(x1)), MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0), MARK(0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true), MARK(true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false), MARK(false))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(true), x1) -> c4(A__ADD(true, mark(x1)), MARK(x1))
   A__ADD(s(false), x1) -> c4(A__ADD(false, mark(x1)), MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2


----------------------------------------

(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c7_2, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1


----------------------------------------

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__PROD(s(z0), z1) -> c7(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), MARK(z1)) by 
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(true, a__prod(mark(x0), mark(true))), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(false, a__prod(mark(x0), mark(false))), MARK(false))

----------------------------------------

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(true, a__prod(mark(x0), mark(true))), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(false, a__prod(mark(x0), mark(false))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(true, a__prod(mark(x0), mark(true))), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(false, a__prod(mark(x0), mark(false))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2


----------------------------------------

(41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__PROD(s(x0), true) -> c7(A__ADD(true, a__prod(mark(x0), mark(true))), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(false, a__prod(mark(x0), mark(false))), MARK(false))

----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)), MARK(0))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)), MARK(true))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2


----------------------------------------

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c8_3, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1


----------------------------------------

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__PROD(s(z0), z1) -> c8(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z0)) by 
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(0))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(true))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(false))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(x0))

----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(0))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(true))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(false))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(x0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(0))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(true))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(false))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(x0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3


----------------------------------------

(47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__PROD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1

Compound Symbols:   c2_1, c9_3, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2


----------------------------------------

(49) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__PROD(s(z0), z1) -> c9(A__ADD(mark(z1), a__prod(mark(z0), mark(z1))), A__PROD(mark(z0), mark(z1)), MARK(z1)) by 
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(false))

----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(false))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(0))
   A__PROD(s(x0), true) -> c9(A__ADD(true, a__prod(mark(x0), mark(true))), A__PROD(mark(x0), mark(true)), MARK(true))
   A__PROD(s(x0), false) -> c9(A__ADD(false, a__prod(mark(x0), mark(false))), A__PROD(mark(x0), mark(false)), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3


----------------------------------------

(51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing tuple parts
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c20_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1


----------------------------------------

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(if(z0, z1, z2)) -> c20(A__IF(mark(z0), z1, z2), MARK(z0)) by 
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(A__IF(s(mark(z0)), x1, x2), MARK(s(z0)))
   MARK(if(0, x1, x2)) -> c20(A__IF(0, x1, x2), MARK(0))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2), MARK(true))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2), MARK(false))

----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(A__IF(s(mark(z0)), x1, x2), MARK(s(z0)))
   MARK(if(0, x1, x2)) -> c20(A__IF(0, x1, x2), MARK(0))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2), MARK(true))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(A__IF(s(mark(z0)), x1, x2), MARK(s(z0)))
   MARK(if(0, x1, x2)) -> c20(A__IF(0, x1, x2), MARK(0))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2), MARK(true))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2


----------------------------------------

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   MARK(if(0, x1, x2)) -> c20(A__IF(0, x1, x2), MARK(0))

----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(A__IF(s(mark(z0)), x1, x2), MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2), MARK(true))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(A__IF(s(mark(z0)), x1, x2), MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2), MARK(true))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2


----------------------------------------

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c22_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1


----------------------------------------

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(prod(z0, z1)) -> c22(A__PROD(mark(z0), mark(z1)), MARK(z0)) by 
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(0, x1)) -> c22(A__PROD(0, mark(x1)), MARK(0))
   MARK(prod(true, x1)) -> c22(A__PROD(true, mark(x1)), MARK(true))
   MARK(prod(false, x1)) -> c22(A__PROD(false, mark(x1)), MARK(false))

----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(0, x1)) -> c22(A__PROD(0, mark(x1)), MARK(0))
   MARK(prod(true, x1)) -> c22(A__PROD(true, mark(x1)), MARK(true))
   MARK(prod(false, x1)) -> c22(A__PROD(false, mark(x1)), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(0, x1)) -> c22(A__PROD(0, mark(x1)), MARK(0))
   MARK(prod(true, x1)) -> c22(A__PROD(true, mark(x1)), MARK(true))
   MARK(prod(false, x1)) -> c22(A__PROD(false, mark(x1)), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2


----------------------------------------

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   MARK(prod(0, x1)) -> c22(A__PROD(0, mark(x1)), MARK(0))
   MARK(prod(false, x1)) -> c22(A__PROD(false, mark(x1)), MARK(false))
   MARK(prod(true, x1)) -> c22(A__PROD(true, mark(x1)), MARK(true))

----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c23_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2


----------------------------------------

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(prod(z0, z1)) -> c23(A__PROD(mark(z0), mark(z1)), MARK(z1)) by 
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0), MARK(0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true), MARK(true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false), MARK(false))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(0, x1)) -> c23(A__PROD(0, mark(x1)), MARK(x1))
   MARK(prod(true, x1)) -> c23(A__PROD(true, mark(x1)), MARK(x1))
   MARK(prod(false, x1)) -> c23(A__PROD(false, mark(x1)), MARK(x1))

----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0), MARK(0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true), MARK(true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false), MARK(false))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(0, x1)) -> c23(A__PROD(0, mark(x1)), MARK(x1))
   MARK(prod(true, x1)) -> c23(A__PROD(true, mark(x1)), MARK(x1))
   MARK(prod(false, x1)) -> c23(A__PROD(false, mark(x1)), MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0), MARK(0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true), MARK(true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false), MARK(false))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(0, x1)) -> c23(A__PROD(0, mark(x1)), MARK(x1))
   MARK(prod(true, x1)) -> c23(A__PROD(true, mark(x1)), MARK(x1))
   MARK(prod(false, x1)) -> c23(A__PROD(false, mark(x1)), MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2


----------------------------------------

(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c24_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1


----------------------------------------

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(p(z0)) -> c24(A__P(mark(z0)), MARK(z0)) by 
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(p(0)) -> c24(A__P(0), MARK(0))
   MARK(p(true)) -> c24(A__P(true), MARK(true))
   MARK(p(false)) -> c24(A__P(false), MARK(false))

----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(p(0)) -> c24(A__P(0), MARK(0))
   MARK(p(true)) -> c24(A__P(true), MARK(true))
   MARK(p(false)) -> c24(A__P(false), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(p(0)) -> c24(A__P(0), MARK(0))
   MARK(p(true)) -> c24(A__P(true), MARK(true))
   MARK(p(false)) -> c24(A__P(false), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2


----------------------------------------

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   MARK(p(false)) -> c24(A__P(false), MARK(false))
   MARK(p(true)) -> c24(A__P(true), MARK(true))
   MARK(p(0)) -> c24(A__P(0), MARK(0))

----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c25_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2


----------------------------------------

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(add(z0, z1)) -> c25(A__ADD(mark(z0), mark(z1)), MARK(z0)) by 
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)), MARK(0))
   MARK(add(true, x1)) -> c25(A__ADD(true, mark(x1)), MARK(true))
   MARK(add(false, x1)) -> c25(A__ADD(false, mark(x1)), MARK(false))

----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)), MARK(0))
   MARK(add(true, x1)) -> c25(A__ADD(true, mark(x1)), MARK(true))
   MARK(add(false, x1)) -> c25(A__ADD(false, mark(x1)), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)), MARK(0))
   MARK(add(true, x1)) -> c25(A__ADD(true, mark(x1)), MARK(true))
   MARK(add(false, x1)) -> c25(A__ADD(false, mark(x1)), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2


----------------------------------------

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   MARK(add(false, x1)) -> c25(A__ADD(false, mark(x1)), MARK(false))
   MARK(add(true, x1)) -> c25(A__ADD(true, mark(x1)), MARK(true))

----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2


----------------------------------------

(75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c26_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1


----------------------------------------

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(add(z0, z1)) -> c26(A__ADD(mark(z0), mark(z1)), MARK(z1)) by 
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0), MARK(0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true), MARK(true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false), MARK(false))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(true, x1)) -> c26(A__ADD(true, mark(x1)), MARK(x1))
   MARK(add(false, x1)) -> c26(A__ADD(false, mark(x1)), MARK(x1))

----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0), MARK(0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true), MARK(true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false), MARK(false))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(true, x1)) -> c26(A__ADD(true, mark(x1)), MARK(x1))
   MARK(add(false, x1)) -> c26(A__ADD(false, mark(x1)), MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0), MARK(0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true), MARK(true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false), MARK(false))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(true, x1)) -> c26(A__ADD(true, mark(x1)), MARK(x1))
   MARK(add(false, x1)) -> c26(A__ADD(false, mark(x1)), MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2


----------------------------------------

(79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__FACT_1, A__PROD_2

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c_2, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1


----------------------------------------

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__FACT(z0) -> c(A__IF(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0)))), MARK(z0)) by 
   A__FACT(x0) -> c(A__IF(zero(mark(x0)), s(0), prod(x0, fact(p(x0)))), MARK(x0))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))), MARK(0))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))), MARK(true))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))), MARK(false))

----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(x0) -> c(A__IF(zero(mark(x0)), s(0), prod(x0, fact(p(x0)))), MARK(x0))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))), MARK(0))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))), MARK(true))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(x0) -> c(A__IF(zero(mark(x0)), s(0), prod(x0, fact(p(x0)))), MARK(x0))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))), MARK(0))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))), MARK(true))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2


----------------------------------------

(83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), fact(z0)) -> c3(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0)) by 
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))), MARK(0))
   A__ADD(s(true), fact(x1)) -> c3(A__ADD(true, a__fact(mark(x1))), MARK(true))
   A__ADD(s(false), fact(x1)) -> c3(A__ADD(false, a__fact(mark(x1))), MARK(false))

----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))), MARK(0))
   A__ADD(s(true), fact(x1)) -> c3(A__ADD(true, a__fact(mark(x1))), MARK(true))
   A__ADD(s(false), fact(x1)) -> c3(A__ADD(false, a__fact(mark(x1))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))), MARK(0))
   A__ADD(s(true), fact(x1)) -> c3(A__ADD(true, a__fact(mark(x1))), MARK(true))
   A__ADD(s(false), fact(x1)) -> c3(A__ADD(false, a__fact(mark(x1))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), fact(x1)) -> c3(A__ADD(true, a__fact(mark(x1))), MARK(true))
   A__ADD(s(false), fact(x1)) -> c3(A__ADD(false, a__fact(mark(x1))), MARK(false))

----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), if(z0, z1, z2)) -> c3(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) by 
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(0))
   A__ADD(s(true), if(x1, x2, x3)) -> c3(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(true))
   A__ADD(s(false), if(x1, x2, x3)) -> c3(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(false))

----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(0))
   A__ADD(s(true), if(x1, x2, x3)) -> c3(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(true))
   A__ADD(s(false), if(x1, x2, x3)) -> c3(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(0))
   A__ADD(s(true), if(x1, x2, x3)) -> c3(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(true))
   A__ADD(s(false), if(x1, x2, x3)) -> c3(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(false), if(x1, x2, x3)) -> c3(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(false))
   A__ADD(s(true), if(x1, x2, x3)) -> c3(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(true))

----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), zero(z0)) -> c3(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0)) by 
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))), MARK(0))
   A__ADD(s(true), zero(x1)) -> c3(A__ADD(true, a__zero(mark(x1))), MARK(true))
   A__ADD(s(false), zero(x1)) -> c3(A__ADD(false, a__zero(mark(x1))), MARK(false))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))), MARK(0))
   A__ADD(s(true), zero(x1)) -> c3(A__ADD(true, a__zero(mark(x1))), MARK(true))
   A__ADD(s(false), zero(x1)) -> c3(A__ADD(false, a__zero(mark(x1))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))), MARK(0))
   A__ADD(s(true), zero(x1)) -> c3(A__ADD(true, a__zero(mark(x1))), MARK(true))
   A__ADD(s(false), zero(x1)) -> c3(A__ADD(false, a__zero(mark(x1))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(false), zero(x1)) -> c3(A__ADD(false, a__zero(mark(x1))), MARK(false))
   A__ADD(s(true), zero(x1)) -> c3(A__ADD(true, a__zero(mark(x1))), MARK(true))

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), prod(z0, z1)) -> c3(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0)) by 
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), prod(x1, x2)) -> c3(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), prod(x1, x2)) -> c3(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(false))

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), prod(x1, x2)) -> c3(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), prod(x1, x2)) -> c3(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), prod(x1, x2)) -> c3(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), prod(x1, x2)) -> c3(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), prod(x1, x2)) -> c3(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), prod(x1, x2)) -> c3(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(false))

----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), p(z0)) -> c3(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0)) by 
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))), MARK(0))
   A__ADD(s(true), p(x1)) -> c3(A__ADD(true, a__p(mark(x1))), MARK(true))
   A__ADD(s(false), p(x1)) -> c3(A__ADD(false, a__p(mark(x1))), MARK(false))

----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))), MARK(0))
   A__ADD(s(true), p(x1)) -> c3(A__ADD(true, a__p(mark(x1))), MARK(true))
   A__ADD(s(false), p(x1)) -> c3(A__ADD(false, a__p(mark(x1))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))), MARK(0))
   A__ADD(s(true), p(x1)) -> c3(A__ADD(true, a__p(mark(x1))), MARK(true))
   A__ADD(s(false), p(x1)) -> c3(A__ADD(false, a__p(mark(x1))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(false), p(x1)) -> c3(A__ADD(false, a__p(mark(x1))), MARK(false))
   A__ADD(s(true), p(x1)) -> c3(A__ADD(true, a__p(mark(x1))), MARK(true))

----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), add(z0, z1)) -> c3(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0)) by 
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), add(x1, x2)) -> c3(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), add(x1, x2)) -> c3(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(false))

----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), add(x1, x2)) -> c3(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), add(x1, x2)) -> c3(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(0))
   A__ADD(s(true), add(x1, x2)) -> c3(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), add(x1, x2)) -> c3(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), add(x1, x2)) -> c3(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(true))
   A__ADD(s(false), add(x1, x2)) -> c3(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(false))

----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), s(z0)) -> c3(A__ADD(mark(x0), s(mark(z0))), MARK(x0)) by 
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))), MARK(0))
   A__ADD(s(true), s(x1)) -> c3(A__ADD(true, s(mark(x1))), MARK(true))
   A__ADD(s(false), s(x1)) -> c3(A__ADD(false, s(mark(x1))), MARK(false))

----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))), MARK(0))
   A__ADD(s(true), s(x1)) -> c3(A__ADD(true, s(mark(x1))), MARK(true))
   A__ADD(s(false), s(x1)) -> c3(A__ADD(false, s(mark(x1))), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))), MARK(0))
   A__ADD(s(true), s(x1)) -> c3(A__ADD(true, s(mark(x1))), MARK(true))
   A__ADD(s(false), s(x1)) -> c3(A__ADD(false, s(mark(x1))), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(false), s(x1)) -> c3(A__ADD(false, s(mark(x1))), MARK(false))
   A__ADD(s(true), s(x1)) -> c3(A__ADD(true, s(mark(x1))), MARK(true))

----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), 0) -> c3(A__ADD(mark(x0), 0), MARK(x0)) by 
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0), MARK(0))
   A__ADD(s(true), 0) -> c3(A__ADD(true, 0), MARK(true))
   A__ADD(s(false), 0) -> c3(A__ADD(false, 0), MARK(false))

----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0), MARK(0))
   A__ADD(s(true), 0) -> c3(A__ADD(true, 0), MARK(true))
   A__ADD(s(false), 0) -> c3(A__ADD(false, 0), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0), MARK(0))
   A__ADD(s(true), 0) -> c3(A__ADD(true, 0), MARK(true))
   A__ADD(s(false), 0) -> c3(A__ADD(false, 0), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), 0) -> c3(A__ADD(true, 0), MARK(true))
   A__ADD(s(false), 0) -> c3(A__ADD(false, 0), MARK(false))

----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), true) -> c3(A__ADD(mark(x0), true), MARK(x0)) by 
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true), MARK(0))
   A__ADD(s(true), true) -> c3(A__ADD(true, true), MARK(true))
   A__ADD(s(false), true) -> c3(A__ADD(false, true), MARK(false))

----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true), MARK(0))
   A__ADD(s(true), true) -> c3(A__ADD(true, true), MARK(true))
   A__ADD(s(false), true) -> c3(A__ADD(false, true), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true), MARK(0))
   A__ADD(s(true), true) -> c3(A__ADD(true, true), MARK(true))
   A__ADD(s(false), true) -> c3(A__ADD(false, true), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(false), true) -> c3(A__ADD(false, true), MARK(false))
   A__ADD(s(true), true) -> c3(A__ADD(true, true), MARK(true))

----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), false) -> c3(A__ADD(mark(x0), false), MARK(x0)) by 
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false), MARK(0))
   A__ADD(s(true), false) -> c3(A__ADD(true, false), MARK(true))
   A__ADD(s(false), false) -> c3(A__ADD(false, false), MARK(false))

----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false), MARK(0))
   A__ADD(s(true), false) -> c3(A__ADD(true, false), MARK(true))
   A__ADD(s(false), false) -> c3(A__ADD(false, false), MARK(false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false), MARK(0))
   A__ADD(s(true), false) -> c3(A__ADD(true, false), MARK(true))
   A__ADD(s(false), false) -> c3(A__ADD(false, false), MARK(false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   A__ADD(s(true), false) -> c3(A__ADD(true, false), MARK(true))
   A__ADD(s(false), false) -> c3(A__ADD(false, false), MARK(false))

----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false), MARK(0))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false), MARK(0))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(fact(z0)), x1) -> c3(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0))) by 
   A__ADD(s(fact(x0)), fact(z0)) -> c3(A__ADD(a__fact(mark(x0)), a__fact(mark(z0))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), if(z0, z1, z2)) -> c3(A__ADD(a__fact(mark(x0)), a__if(mark(z0), z1, z2)), MARK(fact(x0)))
   A__ADD(s(fact(x0)), zero(z0)) -> c3(A__ADD(a__fact(mark(x0)), a__zero(mark(z0))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), prod(z0, z1)) -> c3(A__ADD(a__fact(mark(x0)), a__prod(mark(z0), mark(z1))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), p(z0)) -> c3(A__ADD(a__fact(mark(x0)), a__p(mark(z0))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), add(z0, z1)) -> c3(A__ADD(a__fact(mark(x0)), a__add(mark(z0), mark(z1))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), s(z0)) -> c3(A__ADD(a__fact(mark(x0)), s(mark(z0))), MARK(fact(x0)))
   A__ADD(s(fact(x0)), 0) -> c3(A__ADD(a__fact(mark(x0)), 0), MARK(fact(x0)))
   A__ADD(s(fact(x0)), true) -> c3(A__ADD(a__fact(mark(x0)), true), MARK(fact(x0)))
   A__ADD(s(fact(x0)), false) -> c3(A__ADD(a__fact(mark(x0)), false), MARK(fact(x0)))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(fact(mark(x0)), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))

----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(fact(mark(x0)), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(fact(mark(x0)), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(147) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(if(z0, z1, z2)), x1) -> c3(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) by 
   A__ADD(s(if(x0, x1, x2)), fact(z0)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__fact(mark(z0))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), if(z0, z1, z2)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__if(mark(z0), z1, z2)), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), zero(z0)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__zero(mark(z0))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), prod(z0, z1)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__prod(mark(z0), mark(z1))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), p(z0)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__p(mark(z0))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), add(z0, z1)) -> c3(A__ADD(a__if(mark(x0), x1, x2), a__add(mark(z0), mark(z1))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), s(z0)) -> c3(A__ADD(a__if(mark(x0), x1, x2), s(mark(z0))), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), 0) -> c3(A__ADD(a__if(mark(x0), x1, x2), 0), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), true) -> c3(A__ADD(a__if(mark(x0), x1, x2), true), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, x1, x2)), false) -> c3(A__ADD(a__if(mark(x0), x1, x2), false), MARK(if(x0, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(if(x0, z1, z2)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(if(0, x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))

----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(if(x0, z1, z2)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(if(0, x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(if(x0, z1, z2)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(if(0, x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(153) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(zero(z0)), x1) -> c3(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0))) by 
   A__ADD(s(zero(x0)), fact(z0)) -> c3(A__ADD(a__zero(mark(x0)), a__fact(mark(z0))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), if(z0, z1, z2)) -> c3(A__ADD(a__zero(mark(x0)), a__if(mark(z0), z1, z2)), MARK(zero(x0)))
   A__ADD(s(zero(x0)), zero(z0)) -> c3(A__ADD(a__zero(mark(x0)), a__zero(mark(z0))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), prod(z0, z1)) -> c3(A__ADD(a__zero(mark(x0)), a__prod(mark(z0), mark(z1))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), p(z0)) -> c3(A__ADD(a__zero(mark(x0)), a__p(mark(z0))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), add(z0, z1)) -> c3(A__ADD(a__zero(mark(x0)), a__add(mark(z0), mark(z1))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), s(z0)) -> c3(A__ADD(a__zero(mark(x0)), s(mark(z0))), MARK(zero(x0)))
   A__ADD(s(zero(x0)), 0) -> c3(A__ADD(a__zero(mark(x0)), 0), MARK(zero(x0)))
   A__ADD(s(zero(x0)), true) -> c3(A__ADD(a__zero(mark(x0)), true), MARK(zero(x0)))
   A__ADD(s(zero(x0)), false) -> c3(A__ADD(a__zero(mark(x0)), false), MARK(zero(x0)))
   A__ADD(s(zero(x0)), x1) -> c3(A__ADD(zero(mark(x0)), mark(x1)), MARK(zero(x0)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))

----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(x0)), x1) -> c3(A__ADD(zero(mark(x0)), mark(x1)), MARK(zero(x0)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(x0)), x1) -> c3(A__ADD(zero(mark(x0)), mark(x1)), MARK(zero(x0)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(prod(z0, z1)), x1) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1))) by 
   A__ADD(s(prod(x0, x1)), fact(z0)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__fact(mark(z0))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), if(z0, z1, z2)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__if(mark(z0), z1, z2)), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), zero(z0)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__zero(mark(z0))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), prod(z0, z1)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__prod(mark(z0), mark(z1))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), p(z0)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__p(mark(z0))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), add(z0, z1)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), a__add(mark(z0), mark(z1))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), s(z0)) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), s(mark(z0))), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), 0) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), 0), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), true) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), true), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), false) -> c3(A__ADD(a__prod(mark(x0), mark(x1)), false), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(A__ADD(prod(mark(x0), mark(x1)), mark(x2)), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))

----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(A__ADD(prod(mark(x0), mark(x1)), mark(x2)), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(A__ADD(prod(mark(x0), mark(x1)), mark(x2)), MARK(prod(x0, x1)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(159) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(p(z0)), x1) -> c3(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0))) by 
   A__ADD(s(p(x0)), fact(z0)) -> c3(A__ADD(a__p(mark(x0)), a__fact(mark(z0))), MARK(p(x0)))
   A__ADD(s(p(x0)), if(z0, z1, z2)) -> c3(A__ADD(a__p(mark(x0)), a__if(mark(z0), z1, z2)), MARK(p(x0)))
   A__ADD(s(p(x0)), zero(z0)) -> c3(A__ADD(a__p(mark(x0)), a__zero(mark(z0))), MARK(p(x0)))
   A__ADD(s(p(x0)), prod(z0, z1)) -> c3(A__ADD(a__p(mark(x0)), a__prod(mark(z0), mark(z1))), MARK(p(x0)))
   A__ADD(s(p(x0)), p(z0)) -> c3(A__ADD(a__p(mark(x0)), a__p(mark(z0))), MARK(p(x0)))
   A__ADD(s(p(x0)), add(z0, z1)) -> c3(A__ADD(a__p(mark(x0)), a__add(mark(z0), mark(z1))), MARK(p(x0)))
   A__ADD(s(p(x0)), s(z0)) -> c3(A__ADD(a__p(mark(x0)), s(mark(z0))), MARK(p(x0)))
   A__ADD(s(p(x0)), 0) -> c3(A__ADD(a__p(mark(x0)), 0), MARK(p(x0)))
   A__ADD(s(p(x0)), true) -> c3(A__ADD(a__p(mark(x0)), true), MARK(p(x0)))
   A__ADD(s(p(x0)), false) -> c3(A__ADD(a__p(mark(x0)), false), MARK(p(x0)))
   A__ADD(s(p(x0)), x1) -> c3(A__ADD(p(mark(x0)), mark(x1)), MARK(p(x0)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)), MARK(p(0)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)), MARK(p(true)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)), MARK(p(false)))

----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(x0)), x1) -> c3(A__ADD(p(mark(x0)), mark(x1)), MARK(p(x0)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)), MARK(p(0)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)), MARK(p(true)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)), MARK(p(false)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(x0)), x1) -> c3(A__ADD(p(mark(x0)), mark(x1)), MARK(p(x0)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)), MARK(p(0)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)), MARK(p(true)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)), MARK(p(false)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(165) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(add(z0, z1)), x1) -> c3(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1))) by 
   A__ADD(s(add(x0, x1)), fact(z0)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__fact(mark(z0))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), if(z0, z1, z2)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__if(mark(z0), z1, z2)), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), zero(z0)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__zero(mark(z0))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), prod(z0, z1)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__prod(mark(z0), mark(z1))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), p(z0)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__p(mark(z0))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), add(z0, z1)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), a__add(mark(z0), mark(z1))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), s(z0)) -> c3(A__ADD(a__add(mark(x0), mark(x1)), s(mark(z0))), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), 0) -> c3(A__ADD(a__add(mark(x0), mark(x1)), 0), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), true) -> c3(A__ADD(a__add(mark(x0), mark(x1)), true), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), false) -> c3(A__ADD(a__add(mark(x0), mark(x1)), false), MARK(add(x0, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(A__ADD(add(mark(x0), mark(x1)), mark(x2)), MARK(add(x0, x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))

----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(A__ADD(add(mark(x0), mark(x1)), mark(x2)), MARK(add(x0, x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(A__ADD(add(mark(x0), mark(x1)), mark(x2)), MARK(add(x0, x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_2, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1


----------------------------------------

(169) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(s(z0)), x1) -> c3(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0))) by 
   A__ADD(s(s(x0)), fact(z0)) -> c3(A__ADD(s(mark(x0)), a__fact(mark(z0))), MARK(s(x0)))
   A__ADD(s(s(x0)), if(z0, z1, z2)) -> c3(A__ADD(s(mark(x0)), a__if(mark(z0), z1, z2)), MARK(s(x0)))
   A__ADD(s(s(x0)), zero(z0)) -> c3(A__ADD(s(mark(x0)), a__zero(mark(z0))), MARK(s(x0)))
   A__ADD(s(s(x0)), prod(z0, z1)) -> c3(A__ADD(s(mark(x0)), a__prod(mark(z0), mark(z1))), MARK(s(x0)))
   A__ADD(s(s(x0)), p(z0)) -> c3(A__ADD(s(mark(x0)), a__p(mark(z0))), MARK(s(x0)))
   A__ADD(s(s(x0)), add(z0, z1)) -> c3(A__ADD(s(mark(x0)), a__add(mark(z0), mark(z1))), MARK(s(x0)))
   A__ADD(s(s(x0)), s(z0)) -> c3(A__ADD(s(mark(x0)), s(mark(z0))), MARK(s(x0)))
   A__ADD(s(s(x0)), 0) -> c3(A__ADD(s(mark(x0)), 0), MARK(s(x0)))
   A__ADD(s(s(x0)), true) -> c3(A__ADD(s(mark(x0)), true), MARK(s(x0)))
   A__ADD(s(s(x0)), false) -> c3(A__ADD(s(mark(x0)), false), MARK(s(x0)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))

----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), fact(z0)) -> c4(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0))) by 
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(A__ADD(true, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(A__ADD(false, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))

----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(A__ADD(true, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(A__ADD(false, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(A__ADD(true, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(A__ADD(false, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), if(z0, z1, z2)) -> c4(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) by 
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(if(0, x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))

----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(if(0, x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(if(0, x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(A__ADD(true, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(A__ADD(false, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), zero(z0)) -> c4(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0))) by 
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(A__ADD(true, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(A__ADD(false, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))

----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(A__ADD(true, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(A__ADD(false, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(A__ADD(true, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(A__ADD(false, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(181) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), prod(z0, z1)) -> c4(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1))) by 
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))

----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(A__ADD(true, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(A__ADD(false, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(187) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), p(z0)) -> c4(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0))) by 
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)), MARK(p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)), MARK(p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)), MARK(p(false)))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(true), p(x1)) -> c4(A__ADD(true, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(A__ADD(false, a__p(mark(x1))), MARK(p(x1)))

----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)), MARK(p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)), MARK(p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)), MARK(p(false)))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(true), p(x1)) -> c4(A__ADD(true, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(A__ADD(false, a__p(mark(x1))), MARK(p(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)), MARK(p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)), MARK(p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)), MARK(p(false)))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(true), p(x1)) -> c4(A__ADD(true, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(A__ADD(false, a__p(mark(x1))), MARK(p(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), add(z0, z1)) -> c4(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1))) by 
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))

----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(A__ADD(true, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(A__ADD(false, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(193) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(195) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(x0), s(z0)) -> c4(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0))) by 
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(A__ADD(true, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(A__ADD(false, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))

----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(A__ADD(true, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(A__ADD(false, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(A__ADD(true, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(A__ADD(false, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(199) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(fact(z0)), x1) -> c4(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1)) by 
   A__ADD(s(fact(x0)), fact(z0)) -> c4(A__ADD(a__fact(mark(x0)), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(fact(x0)), if(z0, z1, z2)) -> c4(A__ADD(a__fact(mark(x0)), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(fact(x0)), zero(z0)) -> c4(A__ADD(a__fact(mark(x0)), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(fact(x0)), prod(z0, z1)) -> c4(A__ADD(a__fact(mark(x0)), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(fact(x0)), p(z0)) -> c4(A__ADD(a__fact(mark(x0)), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(fact(x0)), add(z0, z1)) -> c4(A__ADD(a__fact(mark(x0)), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(fact(x0)), s(z0)) -> c4(A__ADD(a__fact(mark(x0)), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0), MARK(0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true), MARK(true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false), MARK(false))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(fact(mark(x0)), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))

----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0), MARK(0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true), MARK(true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false), MARK(false))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(fact(mark(x0)), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0), MARK(0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true), MARK(true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false), MARK(false))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(fact(mark(x0)), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false))
   A__ADD(s(fact(x0)), x1) -> c4(MARK(x1))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false))
   A__ADD(s(fact(x0)), x1) -> c4(MARK(x1))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2


----------------------------------------

(203) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace A__ADD(s(if(z0, z1, z2)), x1) -> c4(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) by 
   A__ADD(s(if(x0, x1, x2)), fact(z0)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__fact(mark(z0))), MARK(fact(z0)))
   A__ADD(s(if(x0, x1, x2)), if(z0, z1, z2)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   A__ADD(s(if(x0, x1, x2)), zero(z0)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__zero(mark(z0))), MARK(zero(z0)))
   A__ADD(s(if(x0, x1, x2)), prod(z0, z1)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   A__ADD(s(if(x0, x1, x2)), p(z0)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__p(mark(z0))), MARK(p(z0)))
   A__ADD(s(if(x0, x1, x2)), add(z0, z1)) -> c4(A__ADD(a__if(mark(x0), x1, x2), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   A__ADD(s(if(x0, x1, x2)), s(z0)) -> c4(A__ADD(a__if(mark(x0), x1, x2), s(mark(z0))), MARK(s(z0)))
   A__ADD(s(if(x0, x1, x2)), 0) -> c4(A__ADD(a__if(mark(x0), x1, x2), 0), MARK(0))
   A__ADD(s(if(x0, x1, x2)), true) -> c4(A__ADD(a__if(mark(x0), x1, x2), true), MARK(true))
   A__ADD(s(if(x0, x1, x2)), false) -> c4(A__ADD(a__if(mark(x0), x1, x2), false), MARK(false))
   A__ADD(s(if(x0, z1, z2)), x3) -> c4(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(x3))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c4(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c4(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(0, x1, x2)), x3) -> c4(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(true, x1, x2)), x3) -> c4(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(false, x1, x2)), x3) -> c4(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(x3))

----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__fact(z0) -> a__if(a__zero(mark(z0)), s(0), prod(z0, fact(p(z0))))
   a__fact(z0) -> fact(z0)
   a__add(0, z0) -> mark(z0)
   a__add(s(z0), z1) -> s(a__add(mark(z0), mark(z1)))
   a__add(z0, z1) -> add(z0, z1)
   a__prod(0, z0) -> 0
   a__prod(s(z0), z1) -> a__add(mark(z1), a__prod(mark(z0), mark(z1)))
   a__prod(z0, z1) -> prod(z0, z1)
   a__if(true, z0, z1) -> mark(z0)
   a__if(false, z0, z1) -> mark(z1)
   a__if(z0, z1, z2) -> if(z0, z1, z2)
   a__zero(0) -> true
   a__zero(s(z0)) -> false
   a__zero(z0) -> zero(z0)
   a__p(s(z0)) -> mark(z0)
   a__p(z0) -> p(z0)
   mark(fact(z0)) -> a__fact(mark(z0))
   mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2)
   mark(zero(z0)) -> a__zero(mark(z0))
   mark(prod(z0, z1)) -> a__prod(mark(z0), mark(z1))
   mark(p(z0)) -> a__p(mark(z0))
   mark(add(z0, z1)) -> a__add(mark(z0), mark(z1))
   mark(s(z0)) -> s(mark(z0))
   mark(0) -> 0
   mark(true) -> true
   mark(false) -> false
Tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false))
   A__ADD(s(fact(x0)), x1) -> c4(MARK(x1))
   A__ADD(s(if(x0, x1, x2)), 0) -> c4(A__ADD(a__if(mark(x0), x1, x2), 0), MARK(0))
   A__ADD(s(if(x0, x1, x2)), true) -> c4(A__ADD(a__if(mark(x0), x1, x2), true), MARK(true))
   A__ADD(s(if(x0, x1, x2)), false) -> c4(A__ADD(a__if(mark(x0), x1, x2), false), MARK(false))
   A__ADD(s(if(x0, z1, z2)), x3) -> c4(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(x3))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c4(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c4(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(0, x1, x2)), x3) -> c4(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(true, x1, x2)), x3) -> c4(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(false, x1, x2)), x3) -> c4(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(x3))
S tuples:
   A__ADD(0, z0) -> c2(MARK(z0))
   A__IF(true, z0, z1) -> c11(MARK(z0))
   A__IF(false, z0, z1) -> c12(MARK(z1))
   A__P(s(z0)) -> c17(MARK(z0))
   MARK(fact(z0)) -> c19(A__FACT(mark(z0)), MARK(z0))
   MARK(s(z0)) -> c27(MARK(z0))
   MARK(zero(z0)) -> c21(MARK(z0))
   A__ADD(s(0), x1) -> c3(A__ADD(0, mark(x1)))
   A__ADD(s(zero(z0)), x1) -> c4(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(prod(z0, z1)), x1) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(p(z0)), x1) -> c4(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(add(z0, z1)), x1) -> c4(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   A__ADD(s(s(z0)), x1) -> c4(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   A__ADD(s(0), x1) -> c4(A__ADD(0, mark(x1)), MARK(x1))
   A__ADD(s(x0), 0) -> c4(A__ADD(mark(x0), 0))
   A__ADD(s(x0), true) -> c4(A__ADD(mark(x0), true))
   A__ADD(s(x0), false) -> c4(A__ADD(mark(x0), false))
   A__ADD(s(true), x1) -> c4(MARK(x1))
   A__ADD(s(false), x1) -> c4(MARK(x1))
   A__PROD(s(x0), x1) -> c7(A__ADD(mark(x1), prod(mark(x0), mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c7(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c7(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c7(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), MARK(x1))
   A__PROD(s(0), x1) -> c7(A__ADD(mark(x1), a__prod(0, mark(x1))), MARK(x1))
   A__PROD(s(true), x1) -> c7(A__ADD(mark(x1), a__prod(true, mark(x1))), MARK(x1))
   A__PROD(s(false), x1) -> c7(A__ADD(mark(x1), a__prod(false, mark(x1))), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c7(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c7(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c7(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c7(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c7(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c7(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c7(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c7(A__ADD(mark(0), a__prod(mark(x0), 0)))
   A__PROD(s(x0), true) -> c7(A__ADD(mark(true), a__prod(mark(x0), true)))
   A__PROD(s(x0), false) -> c7(A__ADD(mark(false), a__prod(mark(x0), false)))
   A__PROD(s(x0), 0) -> c7(A__ADD(0, a__prod(mark(x0), mark(0))))
   A__PROD(s(x0), x1) -> c8(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x0))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(x0), true) -> c8(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(fact(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(fact(z0)))
   A__PROD(s(if(z0, z1, z2)), x1) -> c8(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(if(z0, z1, z2)))
   A__PROD(s(zero(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(zero(z0)))
   A__PROD(s(prod(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(prod(z0, z1)))
   A__PROD(s(p(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(p(z0)))
   A__PROD(s(add(z0, z1)), x1) -> c8(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(add(z0, z1)))
   A__PROD(s(s(z0)), x1) -> c8(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(s(z0)))
   A__PROD(s(x0), fact(z0)) -> c8(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(x0))
   A__PROD(s(x0), if(z0, z1, z2)) -> c8(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(x0))
   A__PROD(s(x0), zero(z0)) -> c8(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(x0))
   A__PROD(s(x0), prod(z0, z1)) -> c8(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(x0))
   A__PROD(s(x0), p(z0)) -> c8(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(x0))
   A__PROD(s(x0), add(z0, z1)) -> c8(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(x0))
   A__PROD(s(x0), s(z0)) -> c8(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(x0))
   A__PROD(s(x0), 0) -> c8(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)), MARK(x0))
   A__PROD(s(0), x1) -> c8(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)))
   A__PROD(s(true), x1) -> c8(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)))
   A__PROD(s(false), x1) -> c8(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)))
   A__PROD(s(x0), true) -> c8(A__PROD(mark(x0), mark(true)), MARK(x0))
   A__PROD(s(x0), false) -> c8(A__PROD(mark(x0), mark(false)), MARK(x0))
   A__PROD(s(x0), x1) -> c9(A__ADD(mark(x1), prod(mark(x0), mark(x1))), A__PROD(mark(x0), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(mark(fact(z0)), a__prod(mark(x0), a__fact(mark(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(mark(if(z0, z1, z2)), a__prod(mark(x0), a__if(mark(z0), z1, z2))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(mark(zero(z0)), a__prod(mark(x0), a__zero(mark(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(mark(prod(z0, z1)), a__prod(mark(x0), a__prod(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(mark(p(z0)), a__prod(mark(x0), a__p(mark(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(mark(add(z0, z1)), a__prod(mark(x0), a__add(mark(z0), mark(z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(mark(s(z0)), a__prod(mark(x0), s(mark(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(fact(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__fact(mark(z0)), mark(x1))), A__PROD(mark(fact(z0)), mark(x1)), MARK(x1))
   A__PROD(s(if(z0, z1, z2)), x1) -> c9(A__ADD(mark(x1), a__prod(a__if(mark(z0), z1, z2), mark(x1))), A__PROD(mark(if(z0, z1, z2)), mark(x1)), MARK(x1))
   A__PROD(s(zero(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__zero(mark(z0)), mark(x1))), A__PROD(mark(zero(z0)), mark(x1)), MARK(x1))
   A__PROD(s(prod(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__prod(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(prod(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(p(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(a__p(mark(z0)), mark(x1))), A__PROD(mark(p(z0)), mark(x1)), MARK(x1))
   A__PROD(s(add(z0, z1)), x1) -> c9(A__ADD(mark(x1), a__prod(a__add(mark(z0), mark(z1)), mark(x1))), A__PROD(mark(add(z0, z1)), mark(x1)), MARK(x1))
   A__PROD(s(s(z0)), x1) -> c9(A__ADD(mark(x1), a__prod(s(mark(z0)), mark(x1))), A__PROD(mark(s(z0)), mark(x1)), MARK(x1))
   A__PROD(s(0), x1) -> c9(A__ADD(mark(x1), a__prod(0, mark(x1))), A__PROD(mark(0), mark(x1)), MARK(x1))
   A__PROD(s(true), x1) -> c9(A__ADD(mark(x1), a__prod(true, mark(x1))), A__PROD(mark(true), mark(x1)), MARK(x1))
   A__PROD(s(false), x1) -> c9(A__ADD(mark(x1), a__prod(false, mark(x1))), A__PROD(mark(false), mark(x1)), MARK(x1))
   A__PROD(s(x0), fact(z0)) -> c9(A__ADD(a__fact(mark(z0)), a__prod(mark(x0), mark(fact(z0)))), A__PROD(mark(x0), mark(fact(z0))), MARK(fact(z0)))
   A__PROD(s(x0), if(z0, z1, z2)) -> c9(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x0), mark(if(z0, z1, z2)))), A__PROD(mark(x0), mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
   A__PROD(s(x0), zero(z0)) -> c9(A__ADD(a__zero(mark(z0)), a__prod(mark(x0), mark(zero(z0)))), A__PROD(mark(x0), mark(zero(z0))), MARK(zero(z0)))
   A__PROD(s(x0), prod(z0, z1)) -> c9(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x0), mark(prod(z0, z1)))), A__PROD(mark(x0), mark(prod(z0, z1))), MARK(prod(z0, z1)))
   A__PROD(s(x0), p(z0)) -> c9(A__ADD(a__p(mark(z0)), a__prod(mark(x0), mark(p(z0)))), A__PROD(mark(x0), mark(p(z0))), MARK(p(z0)))
   A__PROD(s(x0), add(z0, z1)) -> c9(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x0), mark(add(z0, z1)))), A__PROD(mark(x0), mark(add(z0, z1))), MARK(add(z0, z1)))
   A__PROD(s(x0), s(z0)) -> c9(A__ADD(s(mark(z0)), a__prod(mark(x0), mark(s(z0)))), A__PROD(mark(x0), mark(s(z0))), MARK(s(z0)))
   A__PROD(s(x0), 0) -> c9(A__ADD(mark(0), a__prod(mark(x0), 0)), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__ADD(mark(true), a__prod(mark(x0), true)), A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__ADD(mark(false), a__prod(mark(x0), false)), A__PROD(mark(x0), mark(false)))
   A__PROD(s(x0), 0) -> c9(A__ADD(0, a__prod(mark(x0), mark(0))), A__PROD(mark(x0), mark(0)))
   A__PROD(s(x0), true) -> c9(A__PROD(mark(x0), mark(true)))
   A__PROD(s(x0), false) -> c9(A__PROD(mark(x0), mark(false)))
   MARK(if(fact(z0), x1, x2)) -> c20(A__IF(a__fact(mark(z0)), x1, x2), MARK(fact(z0)))
   MARK(if(if(z0, z1, z2), x1, x2)) -> c20(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2)))
   MARK(if(zero(z0), x1, x2)) -> c20(A__IF(a__zero(mark(z0)), x1, x2), MARK(zero(z0)))
   MARK(if(prod(z0, z1), x1, x2)) -> c20(A__IF(a__prod(mark(z0), mark(z1)), x1, x2), MARK(prod(z0, z1)))
   MARK(if(p(z0), x1, x2)) -> c20(A__IF(a__p(mark(z0)), x1, x2), MARK(p(z0)))
   MARK(if(add(z0, z1), x1, x2)) -> c20(A__IF(a__add(mark(z0), mark(z1)), x1, x2), MARK(add(z0, z1)))
   MARK(if(s(z0), x1, x2)) -> c20(MARK(s(z0)))
   MARK(if(true, x1, x2)) -> c20(A__IF(true, x1, x2))
   MARK(if(false, x1, x2)) -> c20(A__IF(false, x1, x2))
   MARK(prod(x0, fact(z0))) -> c22(A__PROD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(prod(x0, if(z0, z1, z2))) -> c22(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(prod(x0, zero(z0))) -> c22(A__PROD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(prod(x0, prod(z0, z1))) -> c22(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, p(z0))) -> c22(A__PROD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(prod(x0, add(z0, z1))) -> c22(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(prod(x0, s(z0))) -> c22(A__PROD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(prod(x0, 0)) -> c22(A__PROD(mark(x0), 0), MARK(x0))
   MARK(prod(x0, true)) -> c22(A__PROD(mark(x0), true), MARK(x0))
   MARK(prod(x0, false)) -> c22(A__PROD(mark(x0), false), MARK(x0))
   MARK(prod(fact(z0), x1)) -> c22(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(prod(if(z0, z1, z2), x1)) -> c22(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(prod(zero(z0), x1)) -> c22(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(prod(prod(z0, z1), x1)) -> c22(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(prod(p(z0), x1)) -> c22(A__PROD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(prod(add(z0, z1), x1)) -> c22(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(prod(s(z0), x1)) -> c22(A__PROD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(prod(x0, fact(z0))) -> c23(A__PROD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(prod(x0, if(z0, z1, z2))) -> c23(A__PROD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(prod(x0, zero(z0))) -> c23(A__PROD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(prod(x0, prod(z0, z1))) -> c23(A__PROD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(prod(x0, p(z0))) -> c23(A__PROD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(prod(x0, add(z0, z1))) -> c23(A__PROD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(prod(x0, s(z0))) -> c23(A__PROD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(prod(fact(z0), x1)) -> c23(A__PROD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(if(z0, z1, z2), x1)) -> c23(A__PROD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(prod(zero(z0), x1)) -> c23(A__PROD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(prod(z0, z1), x1)) -> c23(A__PROD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(p(z0), x1)) -> c23(A__PROD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(add(z0, z1), x1)) -> c23(A__PROD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(prod(s(z0), x1)) -> c23(A__PROD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(prod(x0, 0)) -> c23(A__PROD(mark(x0), 0))
   MARK(prod(x0, true)) -> c23(A__PROD(mark(x0), true))
   MARK(prod(x0, false)) -> c23(A__PROD(mark(x0), false))
   MARK(prod(0, x1)) -> c23(MARK(x1))
   MARK(prod(true, x1)) -> c23(MARK(x1))
   MARK(prod(false, x1)) -> c23(MARK(x1))
   MARK(p(fact(z0))) -> c24(A__P(a__fact(mark(z0))), MARK(fact(z0)))
   MARK(p(if(z0, z1, z2))) -> c24(A__P(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(p(zero(z0))) -> c24(A__P(a__zero(mark(z0))), MARK(zero(z0)))
   MARK(p(prod(z0, z1))) -> c24(A__P(a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(p(p(z0))) -> c24(A__P(a__p(mark(z0))), MARK(p(z0)))
   MARK(p(add(z0, z1))) -> c24(A__P(a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(p(s(z0))) -> c24(A__P(s(mark(z0))), MARK(s(z0)))
   MARK(add(x0, fact(z0))) -> c25(A__ADD(mark(x0), a__fact(mark(z0))), MARK(x0))
   MARK(add(x0, if(z0, z1, z2))) -> c25(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0))
   MARK(add(x0, zero(z0))) -> c25(A__ADD(mark(x0), a__zero(mark(z0))), MARK(x0))
   MARK(add(x0, prod(z0, z1))) -> c25(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, p(z0))) -> c25(A__ADD(mark(x0), a__p(mark(z0))), MARK(x0))
   MARK(add(x0, add(z0, z1))) -> c25(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(x0))
   MARK(add(x0, s(z0))) -> c25(A__ADD(mark(x0), s(mark(z0))), MARK(x0))
   MARK(add(x0, 0)) -> c25(A__ADD(mark(x0), 0), MARK(x0))
   MARK(add(x0, true)) -> c25(A__ADD(mark(x0), true), MARK(x0))
   MARK(add(x0, false)) -> c25(A__ADD(mark(x0), false), MARK(x0))
   MARK(add(fact(z0), x1)) -> c25(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(fact(z0)))
   MARK(add(if(z0, z1, z2), x1)) -> c25(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2)))
   MARK(add(zero(z0), x1)) -> c25(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(zero(z0)))
   MARK(add(prod(z0, z1), x1)) -> c25(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(prod(z0, z1)))
   MARK(add(p(z0), x1)) -> c25(A__ADD(a__p(mark(z0)), mark(x1)), MARK(p(z0)))
   MARK(add(add(z0, z1), x1)) -> c25(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(add(z0, z1)))
   MARK(add(s(z0), x1)) -> c25(A__ADD(s(mark(z0)), mark(x1)), MARK(s(z0)))
   MARK(add(0, x1)) -> c25(A__ADD(0, mark(x1)))
   MARK(add(x0, fact(z0))) -> c26(A__ADD(mark(x0), a__fact(mark(z0))), MARK(fact(z0)))
   MARK(add(x0, if(z0, z1, z2))) -> c26(A__ADD(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2)))
   MARK(add(x0, zero(z0))) -> c26(A__ADD(mark(x0), a__zero(mark(z0))), MARK(zero(z0)))
   MARK(add(x0, prod(z0, z1))) -> c26(A__ADD(mark(x0), a__prod(mark(z0), mark(z1))), MARK(prod(z0, z1)))
   MARK(add(x0, p(z0))) -> c26(A__ADD(mark(x0), a__p(mark(z0))), MARK(p(z0)))
   MARK(add(x0, add(z0, z1))) -> c26(A__ADD(mark(x0), a__add(mark(z0), mark(z1))), MARK(add(z0, z1)))
   MARK(add(x0, s(z0))) -> c26(A__ADD(mark(x0), s(mark(z0))), MARK(s(z0)))
   MARK(add(fact(z0), x1)) -> c26(A__ADD(a__fact(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(if(z0, z1, z2), x1)) -> c26(A__ADD(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1))
   MARK(add(zero(z0), x1)) -> c26(A__ADD(a__zero(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(prod(z0, z1), x1)) -> c26(A__ADD(a__prod(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(p(z0), x1)) -> c26(A__ADD(a__p(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(add(z0, z1), x1)) -> c26(A__ADD(a__add(mark(z0), mark(z1)), mark(x1)), MARK(x1))
   MARK(add(s(z0), x1)) -> c26(A__ADD(s(mark(z0)), mark(x1)), MARK(x1))
   MARK(add(0, x1)) -> c26(A__ADD(0, mark(x1)), MARK(x1))
   MARK(add(x0, 0)) -> c26(A__ADD(mark(x0), 0))
   MARK(add(x0, true)) -> c26(A__ADD(mark(x0), true))
   MARK(add(x0, false)) -> c26(A__ADD(mark(x0), false))
   MARK(add(true, x1)) -> c26(MARK(x1))
   MARK(add(false, x1)) -> c26(MARK(x1))
   A__FACT(fact(z0)) -> c(A__IF(a__zero(a__fact(mark(z0))), s(0), prod(fact(z0), fact(p(fact(z0))))), MARK(fact(z0)))
   A__FACT(if(z0, z1, z2)) -> c(A__IF(a__zero(a__if(mark(z0), z1, z2)), s(0), prod(if(z0, z1, z2), fact(p(if(z0, z1, z2))))), MARK(if(z0, z1, z2)))
   A__FACT(zero(z0)) -> c(A__IF(a__zero(a__zero(mark(z0))), s(0), prod(zero(z0), fact(p(zero(z0))))), MARK(zero(z0)))
   A__FACT(prod(z0, z1)) -> c(A__IF(a__zero(a__prod(mark(z0), mark(z1))), s(0), prod(prod(z0, z1), fact(p(prod(z0, z1))))), MARK(prod(z0, z1)))
   A__FACT(p(z0)) -> c(A__IF(a__zero(a__p(mark(z0))), s(0), prod(p(z0), fact(p(p(z0))))), MARK(p(z0)))
   A__FACT(add(z0, z1)) -> c(A__IF(a__zero(a__add(mark(z0), mark(z1))), s(0), prod(add(z0, z1), fact(p(add(z0, z1))))), MARK(add(z0, z1)))
   A__FACT(s(z0)) -> c(A__IF(a__zero(s(mark(z0))), s(0), prod(s(z0), fact(p(s(z0))))), MARK(s(z0)))
   A__FACT(x0) -> c(MARK(x0))
   A__FACT(0) -> c(A__IF(a__zero(0), s(0), prod(0, fact(p(0)))))
   A__FACT(true) -> c(A__IF(a__zero(true), s(0), prod(true, fact(p(true)))))
   A__FACT(false) -> c(A__IF(a__zero(false), s(0), prod(false, fact(p(false)))))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(x0))
   A__ADD(s(x0), fact(x1)) -> c3(A__ADD(mark(x0), fact(mark(x1))), MARK(x0))
   A__ADD(s(x0), fact(fact(z0))) -> c3(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), fact(zero(z0))) -> c3(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(p(z0))) -> c3(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(add(z0, z1))) -> c3(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), fact(s(z0))) -> c3(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), fact(0)) -> c3(A__ADD(mark(x0), a__fact(0)), MARK(x0))
   A__ADD(s(x0), fact(true)) -> c3(A__ADD(mark(x0), a__fact(true)), MARK(x0))
   A__ADD(s(x0), fact(false)) -> c3(A__ADD(mark(x0), a__fact(false)), MARK(x0))
   A__ADD(s(fact(z0)), fact(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), fact(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), fact(x1)) -> c3(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), fact(x1)) -> c3(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), fact(x1)) -> c3(A__ADD(0, a__fact(mark(x1))))
   A__ADD(s(x0), if(x1, z1, z2)) -> c3(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(x0))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c3(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c3(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0))
   A__ADD(s(x0), if(0, x2, x3)) -> c3(A__ADD(mark(x0), a__if(0, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(true, x2, x3)) -> c3(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(x0))
   A__ADD(s(x0), if(false, x2, x3)) -> c3(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(x0))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c3(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c3(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0)))
   A__ADD(s(0), if(x1, x2, x3)) -> c3(A__ADD(0, a__if(mark(x1), x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c3(A__ADD(mark(x0), zero(mark(x1))), MARK(x0))
   A__ADD(s(x0), zero(fact(z0))) -> c3(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), zero(zero(z0))) -> c3(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(p(z0))) -> c3(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(add(z0, z1))) -> c3(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), zero(s(z0))) -> c3(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), zero(0)) -> c3(A__ADD(mark(x0), a__zero(0)), MARK(x0))
   A__ADD(s(x0), zero(true)) -> c3(A__ADD(mark(x0), a__zero(true)), MARK(x0))
   A__ADD(s(x0), zero(false)) -> c3(A__ADD(mark(x0), a__zero(false)), MARK(x0))
   A__ADD(s(fact(z0)), zero(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), zero(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), zero(x1)) -> c3(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), zero(x1)) -> c3(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), zero(x1)) -> c3(A__ADD(0, a__zero(mark(x1))))
   A__ADD(s(x0), prod(x1, x2)) -> c3(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, p(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), prod(x1, s(z0))) -> c3(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), prod(x1, 0)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), prod(x1, true)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), prod(x1, false)) -> c3(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(p(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(s(z0), x2)) -> c3(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(0, x2)) -> c3(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(true, x2)) -> c3(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), prod(false, x2)) -> c3(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), prod(x1, x2)) -> c3(A__ADD(0, a__prod(mark(x1), mark(x2))))
   A__ADD(s(x0), p(x1)) -> c3(A__ADD(mark(x0), p(mark(x1))), MARK(x0))
   A__ADD(s(x0), p(fact(z0))) -> c3(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), p(zero(z0))) -> c3(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(prod(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(p(z0))) -> c3(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(add(z0, z1))) -> c3(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), p(s(z0))) -> c3(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), p(0)) -> c3(A__ADD(mark(x0), a__p(0)), MARK(x0))
   A__ADD(s(x0), p(true)) -> c3(A__ADD(mark(x0), a__p(true)), MARK(x0))
   A__ADD(s(x0), p(false)) -> c3(A__ADD(mark(x0), a__p(false)), MARK(x0))
   A__ADD(s(fact(z0)), p(x1)) -> c3(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), p(x1)) -> c3(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), p(x1)) -> c3(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), p(x1)) -> c3(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), p(x1)) -> c3(A__ADD(0, a__p(mark(x1))))
   A__ADD(s(x0), add(x1, x2)) -> c3(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(x1, fact(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), add(x1, zero(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, p(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c3(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), add(x1, s(z0))) -> c3(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), add(x1, 0)) -> c3(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(x0))
   A__ADD(s(x0), add(x1, true)) -> c3(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(x0))
   A__ADD(s(x0), add(x1, false)) -> c3(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(x0))
   A__ADD(s(x0), add(fact(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c3(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(zero(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(p(z0), x2)) -> c3(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c3(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(s(z0), x2)) -> c3(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(x0))
   A__ADD(s(x0), add(0, x2)) -> c3(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(true, x2)) -> c3(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(x0))
   A__ADD(s(x0), add(false, x2)) -> c3(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(x0))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c3(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c3(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c3(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c3(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c3(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(s(z0)))
   A__ADD(s(0), add(x1, x2)) -> c3(A__ADD(0, a__add(mark(x1), mark(x2))))
   A__ADD(s(x0), s(fact(z0))) -> c3(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c3(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(x0))
   A__ADD(s(x0), s(zero(z0))) -> c3(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(prod(z0, z1))) -> c3(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(p(z0))) -> c3(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(add(z0, z1))) -> c3(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(x0))
   A__ADD(s(x0), s(s(z0))) -> c3(A__ADD(mark(x0), s(s(mark(z0)))), MARK(x0))
   A__ADD(s(x0), s(0)) -> c3(A__ADD(mark(x0), s(0)), MARK(x0))
   A__ADD(s(x0), s(true)) -> c3(A__ADD(mark(x0), s(true)), MARK(x0))
   A__ADD(s(x0), s(false)) -> c3(A__ADD(mark(x0), s(false)), MARK(x0))
   A__ADD(s(fact(z0)), s(x1)) -> c3(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c3(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), s(x1)) -> c3(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), s(x1)) -> c3(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c3(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), s(x1)) -> c3(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(z0)))
   A__ADD(s(0), s(x1)) -> c3(A__ADD(0, s(mark(x1))))
   A__ADD(s(fact(z0)), 0) -> c3(A__ADD(a__fact(mark(z0)), 0), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), 0) -> c3(A__ADD(a__if(mark(z0), z1, z2), 0), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), 0) -> c3(A__ADD(a__zero(mark(z0)), 0), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), 0) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), 0), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), 0) -> c3(A__ADD(a__p(mark(z0)), 0), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), 0) -> c3(A__ADD(a__add(mark(z0), mark(z1)), 0), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), 0) -> c3(A__ADD(s(mark(z0)), 0), MARK(s(z0)))
   A__ADD(s(0), 0) -> c3(A__ADD(0, 0))
   A__ADD(s(fact(z0)), true) -> c3(A__ADD(a__fact(mark(z0)), true), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), true) -> c3(A__ADD(a__if(mark(z0), z1, z2), true), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), true) -> c3(A__ADD(a__zero(mark(z0)), true), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), true) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), true), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), true) -> c3(A__ADD(a__p(mark(z0)), true), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), true) -> c3(A__ADD(a__add(mark(z0), mark(z1)), true), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), true) -> c3(A__ADD(s(mark(z0)), true), MARK(s(z0)))
   A__ADD(s(0), true) -> c3(A__ADD(0, true))
   A__ADD(s(fact(z0)), false) -> c3(A__ADD(a__fact(mark(z0)), false), MARK(fact(z0)))
   A__ADD(s(if(z0, z1, z2)), false) -> c3(A__ADD(a__if(mark(z0), z1, z2), false), MARK(if(z0, z1, z2)))
   A__ADD(s(zero(z0)), false) -> c3(A__ADD(a__zero(mark(z0)), false), MARK(zero(z0)))
   A__ADD(s(prod(z0, z1)), false) -> c3(A__ADD(a__prod(mark(z0), mark(z1)), false), MARK(prod(z0, z1)))
   A__ADD(s(p(z0)), false) -> c3(A__ADD(a__p(mark(z0)), false), MARK(p(z0)))
   A__ADD(s(add(z0, z1)), false) -> c3(A__ADD(a__add(mark(z0), mark(z1)), false), MARK(add(z0, z1)))
   A__ADD(s(s(z0)), false) -> c3(A__ADD(s(mark(z0)), false), MARK(s(z0)))
   A__ADD(s(0), false) -> c3(A__ADD(0, false))
   A__ADD(s(fact(x0)), x1) -> c3(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(fact(x0)))
   A__ADD(s(fact(fact(z0))), x1) -> c3(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(fact(fact(z0))))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c3(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(fact(zero(z0))), x1) -> c3(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(fact(zero(z0))))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c3(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(fact(prod(z0, z1))))
   A__ADD(s(fact(p(z0))), x1) -> c3(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(fact(p(z0))))
   A__ADD(s(fact(add(z0, z1))), x1) -> c3(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(fact(add(z0, z1))))
   A__ADD(s(fact(s(z0))), x1) -> c3(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(fact(s(z0))))
   A__ADD(s(fact(0)), x1) -> c3(A__ADD(a__fact(0), mark(x1)), MARK(fact(0)))
   A__ADD(s(fact(true)), x1) -> c3(A__ADD(a__fact(true), mark(x1)), MARK(fact(true)))
   A__ADD(s(fact(false)), x1) -> c3(A__ADD(a__fact(false), mark(x1)), MARK(fact(false)))
   A__ADD(s(fact(x0)), x1) -> c3(MARK(fact(x0)))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(if(fact(z0), x1, x2)))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c3(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(if(if(z0, z1, z2), x1, x2)))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(if(zero(z0), x1, x2)))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(prod(z0, z1), x1, x2)))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c3(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(if(p(z0), x1, x2)))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c3(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(if(add(z0, z1), x1, x2)))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c3(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(if(s(z0), x1, x2)))
   A__ADD(s(if(true, x1, x2)), x3) -> c3(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(if(true, x1, x2)))
   A__ADD(s(if(false, x1, x2)), x3) -> c3(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(if(false, x1, x2)))
   A__ADD(s(if(x0, z1, z2)), x3) -> c3(MARK(if(x0, z1, z2)))
   A__ADD(s(if(0, x1, x2)), x3) -> c3(A__ADD(a__if(0, x1, x2), mark(x3)))
   A__ADD(s(zero(fact(z0))), x1) -> c3(A__ADD(a__zero(a__fact(mark(z0))), mark(x1)), MARK(zero(fact(z0))))
   A__ADD(s(zero(if(z0, z1, z2))), x1) -> c3(A__ADD(a__zero(a__if(mark(z0), z1, z2)), mark(x1)), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(zero(zero(z0))), x1) -> c3(A__ADD(a__zero(a__zero(mark(z0))), mark(x1)), MARK(zero(zero(z0))))
   A__ADD(s(zero(prod(z0, z1))), x1) -> c3(A__ADD(a__zero(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(zero(prod(z0, z1))))
   A__ADD(s(zero(p(z0))), x1) -> c3(A__ADD(a__zero(a__p(mark(z0))), mark(x1)), MARK(zero(p(z0))))
   A__ADD(s(zero(add(z0, z1))), x1) -> c3(A__ADD(a__zero(a__add(mark(z0), mark(z1))), mark(x1)), MARK(zero(add(z0, z1))))
   A__ADD(s(zero(s(z0))), x1) -> c3(A__ADD(a__zero(s(mark(z0))), mark(x1)), MARK(zero(s(z0))))
   A__ADD(s(zero(0)), x1) -> c3(A__ADD(a__zero(0), mark(x1)), MARK(zero(0)))
   A__ADD(s(zero(true)), x1) -> c3(A__ADD(a__zero(true), mark(x1)), MARK(zero(true)))
   A__ADD(s(zero(false)), x1) -> c3(A__ADD(a__zero(false), mark(x1)), MARK(zero(false)))
   A__ADD(s(zero(x0)), x1) -> c3(MARK(zero(x0)))
   A__ADD(s(prod(x0, fact(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(prod(x0, fact(z0))))
   A__ADD(s(prod(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__prod(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(prod(x0, if(z0, z1, z2))))
   A__ADD(s(prod(x0, zero(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(prod(x0, zero(z0))))
   A__ADD(s(prod(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, prod(z0, z1))))
   A__ADD(s(prod(x0, p(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), a__p(mark(z0))), mark(x2)), MARK(prod(x0, p(z0))))
   A__ADD(s(prod(x0, add(z0, z1))), x2) -> c3(A__ADD(a__prod(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(prod(x0, add(z0, z1))))
   A__ADD(s(prod(x0, s(z0))), x2) -> c3(A__ADD(a__prod(mark(x0), s(mark(z0))), mark(x2)), MARK(prod(x0, s(z0))))
   A__ADD(s(prod(x0, 0)), x2) -> c3(A__ADD(a__prod(mark(x0), 0), mark(x2)), MARK(prod(x0, 0)))
   A__ADD(s(prod(x0, true)), x2) -> c3(A__ADD(a__prod(mark(x0), true), mark(x2)), MARK(prod(x0, true)))
   A__ADD(s(prod(x0, false)), x2) -> c3(A__ADD(a__prod(mark(x0), false), mark(x2)), MARK(prod(x0, false)))
   A__ADD(s(prod(fact(z0), x1)), x2) -> c3(A__ADD(a__prod(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(prod(fact(z0), x1)))
   A__ADD(s(prod(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__prod(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(prod(if(z0, z1, z2), x1)))
   A__ADD(s(prod(zero(z0), x1)), x2) -> c3(A__ADD(a__prod(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(prod(zero(z0), x1)))
   A__ADD(s(prod(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(prod(z0, z1), x1)))
   A__ADD(s(prod(p(z0), x1)), x2) -> c3(A__ADD(a__prod(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(prod(p(z0), x1)))
   A__ADD(s(prod(add(z0, z1), x1)), x2) -> c3(A__ADD(a__prod(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(prod(add(z0, z1), x1)))
   A__ADD(s(prod(s(z0), x1)), x2) -> c3(A__ADD(a__prod(s(mark(z0)), mark(x1)), mark(x2)), MARK(prod(s(z0), x1)))
   A__ADD(s(prod(0, x1)), x2) -> c3(A__ADD(a__prod(0, mark(x1)), mark(x2)), MARK(prod(0, x1)))
   A__ADD(s(prod(true, x1)), x2) -> c3(A__ADD(a__prod(true, mark(x1)), mark(x2)), MARK(prod(true, x1)))
   A__ADD(s(prod(false, x1)), x2) -> c3(A__ADD(a__prod(false, mark(x1)), mark(x2)), MARK(prod(false, x1)))
   A__ADD(s(prod(x0, x1)), x2) -> c3(MARK(prod(x0, x1)))
   A__ADD(s(p(fact(z0))), x1) -> c3(A__ADD(a__p(a__fact(mark(z0))), mark(x1)), MARK(p(fact(z0))))
   A__ADD(s(p(if(z0, z1, z2))), x1) -> c3(A__ADD(a__p(a__if(mark(z0), z1, z2)), mark(x1)), MARK(p(if(z0, z1, z2))))
   A__ADD(s(p(zero(z0))), x1) -> c3(A__ADD(a__p(a__zero(mark(z0))), mark(x1)), MARK(p(zero(z0))))
   A__ADD(s(p(prod(z0, z1))), x1) -> c3(A__ADD(a__p(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(p(prod(z0, z1))))
   A__ADD(s(p(p(z0))), x1) -> c3(A__ADD(a__p(a__p(mark(z0))), mark(x1)), MARK(p(p(z0))))
   A__ADD(s(p(add(z0, z1))), x1) -> c3(A__ADD(a__p(a__add(mark(z0), mark(z1))), mark(x1)), MARK(p(add(z0, z1))))
   A__ADD(s(p(s(z0))), x1) -> c3(A__ADD(a__p(s(mark(z0))), mark(x1)), MARK(p(s(z0))))
   A__ADD(s(p(x0)), x1) -> c3(MARK(p(x0)))
   A__ADD(s(p(0)), x1) -> c3(A__ADD(a__p(0), mark(x1)))
   A__ADD(s(p(true)), x1) -> c3(A__ADD(a__p(true), mark(x1)))
   A__ADD(s(p(false)), x1) -> c3(A__ADD(a__p(false), mark(x1)))
   A__ADD(s(add(x0, fact(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__fact(mark(z0))), mark(x2)), MARK(add(x0, fact(z0))))
   A__ADD(s(add(x0, if(z0, z1, z2))), x2) -> c3(A__ADD(a__add(mark(x0), a__if(mark(z0), z1, z2)), mark(x2)), MARK(add(x0, if(z0, z1, z2))))
   A__ADD(s(add(x0, zero(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__zero(mark(z0))), mark(x2)), MARK(add(x0, zero(z0))))
   A__ADD(s(add(x0, prod(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__prod(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, prod(z0, z1))))
   A__ADD(s(add(x0, p(z0))), x2) -> c3(A__ADD(a__add(mark(x0), a__p(mark(z0))), mark(x2)), MARK(add(x0, p(z0))))
   A__ADD(s(add(x0, add(z0, z1))), x2) -> c3(A__ADD(a__add(mark(x0), a__add(mark(z0), mark(z1))), mark(x2)), MARK(add(x0, add(z0, z1))))
   A__ADD(s(add(x0, s(z0))), x2) -> c3(A__ADD(a__add(mark(x0), s(mark(z0))), mark(x2)), MARK(add(x0, s(z0))))
   A__ADD(s(add(x0, 0)), x2) -> c3(A__ADD(a__add(mark(x0), 0), mark(x2)), MARK(add(x0, 0)))
   A__ADD(s(add(x0, true)), x2) -> c3(A__ADD(a__add(mark(x0), true), mark(x2)), MARK(add(x0, true)))
   A__ADD(s(add(x0, false)), x2) -> c3(A__ADD(a__add(mark(x0), false), mark(x2)), MARK(add(x0, false)))
   A__ADD(s(add(fact(z0), x1)), x2) -> c3(A__ADD(a__add(a__fact(mark(z0)), mark(x1)), mark(x2)), MARK(add(fact(z0), x1)))
   A__ADD(s(add(if(z0, z1, z2), x1)), x2) -> c3(A__ADD(a__add(a__if(mark(z0), z1, z2), mark(x1)), mark(x2)), MARK(add(if(z0, z1, z2), x1)))
   A__ADD(s(add(zero(z0), x1)), x2) -> c3(A__ADD(a__add(a__zero(mark(z0)), mark(x1)), mark(x2)), MARK(add(zero(z0), x1)))
   A__ADD(s(add(prod(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__prod(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(prod(z0, z1), x1)))
   A__ADD(s(add(p(z0), x1)), x2) -> c3(A__ADD(a__add(a__p(mark(z0)), mark(x1)), mark(x2)), MARK(add(p(z0), x1)))
   A__ADD(s(add(add(z0, z1), x1)), x2) -> c3(A__ADD(a__add(a__add(mark(z0), mark(z1)), mark(x1)), mark(x2)), MARK(add(add(z0, z1), x1)))
   A__ADD(s(add(s(z0), x1)), x2) -> c3(A__ADD(a__add(s(mark(z0)), mark(x1)), mark(x2)), MARK(add(s(z0), x1)))
   A__ADD(s(add(0, x1)), x2) -> c3(A__ADD(a__add(0, mark(x1)), mark(x2)), MARK(add(0, x1)))
   A__ADD(s(add(true, x1)), x2) -> c3(A__ADD(a__add(true, mark(x1)), mark(x2)), MARK(add(true, x1)))
   A__ADD(s(add(false, x1)), x2) -> c3(A__ADD(a__add(false, mark(x1)), mark(x2)), MARK(add(false, x1)))
   A__ADD(s(add(x0, x1)), x2) -> c3(MARK(add(x0, x1)))
   A__ADD(s(s(fact(z0))), x1) -> c3(A__ADD(s(a__fact(mark(z0))), mark(x1)), MARK(s(fact(z0))))
   A__ADD(s(s(if(z0, z1, z2))), x1) -> c3(A__ADD(s(a__if(mark(z0), z1, z2)), mark(x1)), MARK(s(if(z0, z1, z2))))
   A__ADD(s(s(zero(z0))), x1) -> c3(A__ADD(s(a__zero(mark(z0))), mark(x1)), MARK(s(zero(z0))))
   A__ADD(s(s(prod(z0, z1))), x1) -> c3(A__ADD(s(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(s(prod(z0, z1))))
   A__ADD(s(s(p(z0))), x1) -> c3(A__ADD(s(a__p(mark(z0))), mark(x1)), MARK(s(p(z0))))
   A__ADD(s(s(add(z0, z1))), x1) -> c3(A__ADD(s(a__add(mark(z0), mark(z1))), mark(x1)), MARK(s(add(z0, z1))))
   A__ADD(s(s(s(z0))), x1) -> c3(A__ADD(s(s(mark(z0))), mark(x1)), MARK(s(s(z0))))
   A__ADD(s(s(0)), x1) -> c3(A__ADD(s(0), mark(x1)), MARK(s(0)))
   A__ADD(s(s(true)), x1) -> c3(A__ADD(s(true), mark(x1)), MARK(s(true)))
   A__ADD(s(s(false)), x1) -> c3(A__ADD(s(false), mark(x1)), MARK(s(false)))
   A__ADD(s(s(x0)), x1) -> c3(MARK(s(x0)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(mark(x1))), s(0), prod(mark(x1), fact(p(mark(x1)))))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(A__ADD(mark(x0), fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(fact(z0))) -> c4(A__ADD(mark(x0), a__fact(a__fact(mark(z0)))), MARK(fact(fact(z0))))
   A__ADD(s(x0), fact(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__fact(a__if(mark(z0), z1, z2))), MARK(fact(if(z0, z1, z2))))
   A__ADD(s(x0), fact(zero(z0))) -> c4(A__ADD(mark(x0), a__fact(a__zero(mark(z0)))), MARK(fact(zero(z0))))
   A__ADD(s(x0), fact(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__prod(mark(z0), mark(z1)))), MARK(fact(prod(z0, z1))))
   A__ADD(s(x0), fact(p(z0))) -> c4(A__ADD(mark(x0), a__fact(a__p(mark(z0)))), MARK(fact(p(z0))))
   A__ADD(s(x0), fact(add(z0, z1))) -> c4(A__ADD(mark(x0), a__fact(a__add(mark(z0), mark(z1)))), MARK(fact(add(z0, z1))))
   A__ADD(s(x0), fact(s(z0))) -> c4(A__ADD(mark(x0), a__fact(s(mark(z0)))), MARK(fact(s(z0))))
   A__ADD(s(x0), fact(0)) -> c4(A__ADD(mark(x0), a__fact(0)), MARK(fact(0)))
   A__ADD(s(x0), fact(true)) -> c4(A__ADD(mark(x0), a__fact(true)), MARK(fact(true)))
   A__ADD(s(x0), fact(false)) -> c4(A__ADD(mark(x0), a__fact(false)), MARK(fact(false)))
   A__ADD(s(fact(z0)), fact(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(if(z0, z1, z2)), fact(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(zero(z0)), fact(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(prod(z0, z1)), fact(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(p(z0)), fact(x1)) -> c4(A__ADD(a__p(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(add(z0, z1)), fact(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(s(z0)), fact(x1)) -> c4(A__ADD(s(mark(z0)), a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(0), fact(x1)) -> c4(A__ADD(0, a__fact(mark(x1))), MARK(fact(x1)))
   A__ADD(s(x0), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(true), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(false), fact(x1)) -> c4(MARK(fact(x1)))
   A__ADD(s(x0), if(x1, z1, z2)) -> c4(A__ADD(mark(x0), if(mark(x1), z1, z2)), MARK(if(x1, z1, z2)))
   A__ADD(s(x0), if(fact(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__fact(mark(z0)), x2, x3)), MARK(if(fact(z0), x2, x3)))
   A__ADD(s(x0), if(if(z0, z1, z2), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(if(if(z0, z1, z2), x2, x3)))
   A__ADD(s(x0), if(zero(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__zero(mark(z0)), x2, x3)), MARK(if(zero(z0), x2, x3)))
   A__ADD(s(x0), if(prod(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__prod(mark(z0), mark(z1)), x2, x3)), MARK(if(prod(z0, z1), x2, x3)))
   A__ADD(s(x0), if(p(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__p(mark(z0)), x2, x3)), MARK(if(p(z0), x2, x3)))
   A__ADD(s(x0), if(add(z0, z1), x2, x3)) -> c4(A__ADD(mark(x0), a__if(a__add(mark(z0), mark(z1)), x2, x3)), MARK(if(add(z0, z1), x2, x3)))
   A__ADD(s(x0), if(s(z0), x2, x3)) -> c4(A__ADD(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(if(s(z0), x2, x3)))
   A__ADD(s(x0), if(true, x2, x3)) -> c4(A__ADD(mark(x0), a__if(true, x2, x3)), MARK(if(true, x2, x3)))
   A__ADD(s(x0), if(false, x2, x3)) -> c4(A__ADD(mark(x0), a__if(false, x2, x3)), MARK(if(false, x2, x3)))
   A__ADD(s(fact(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__fact(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(if(z0, z1, z2)), if(x1, x2, x3)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(zero(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__zero(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(prod(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(p(z0)), if(x1, x2, x3)) -> c4(A__ADD(a__p(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(add(z0, z1)), if(x1, x2, x3)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(s(z0)), if(x1, x2, x3)) -> c4(A__ADD(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(0), if(x1, x2, x3)) -> c4(A__ADD(0, a__if(mark(x1), x2, x3)), MARK(if(x1, x2, x3)))
   A__ADD(s(x0), if(0, x2, x3)) -> c4(A__ADD(mark(x0), a__if(0, x2, x3)))
   A__ADD(s(true), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(false), if(x1, x2, x3)) -> c4(MARK(if(x1, x2, x3)))
   A__ADD(s(x0), zero(x1)) -> c4(A__ADD(mark(x0), zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(fact(z0))) -> c4(A__ADD(mark(x0), a__zero(a__fact(mark(z0)))), MARK(zero(fact(z0))))
   A__ADD(s(x0), zero(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__zero(a__if(mark(z0), z1, z2))), MARK(zero(if(z0, z1, z2))))
   A__ADD(s(x0), zero(zero(z0))) -> c4(A__ADD(mark(x0), a__zero(a__zero(mark(z0)))), MARK(zero(zero(z0))))
   A__ADD(s(x0), zero(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__prod(mark(z0), mark(z1)))), MARK(zero(prod(z0, z1))))
   A__ADD(s(x0), zero(p(z0))) -> c4(A__ADD(mark(x0), a__zero(a__p(mark(z0)))), MARK(zero(p(z0))))
   A__ADD(s(x0), zero(add(z0, z1))) -> c4(A__ADD(mark(x0), a__zero(a__add(mark(z0), mark(z1)))), MARK(zero(add(z0, z1))))
   A__ADD(s(x0), zero(s(z0))) -> c4(A__ADD(mark(x0), a__zero(s(mark(z0)))), MARK(zero(s(z0))))
   A__ADD(s(x0), zero(0)) -> c4(A__ADD(mark(x0), a__zero(0)), MARK(zero(0)))
   A__ADD(s(x0), zero(true)) -> c4(A__ADD(mark(x0), a__zero(true)), MARK(zero(true)))
   A__ADD(s(x0), zero(false)) -> c4(A__ADD(mark(x0), a__zero(false)), MARK(zero(false)))
   A__ADD(s(fact(z0)), zero(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(if(z0, z1, z2)), zero(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(zero(z0)), zero(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(prod(z0, z1)), zero(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(p(z0)), zero(x1)) -> c4(A__ADD(a__p(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(add(z0, z1)), zero(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(s(z0)), zero(x1)) -> c4(A__ADD(s(mark(z0)), a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(0), zero(x1)) -> c4(A__ADD(0, a__zero(mark(x1))), MARK(zero(x1)))
   A__ADD(s(x0), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(true), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(false), zero(x1)) -> c4(MARK(zero(x1)))
   A__ADD(s(x0), prod(x1, x2)) -> c4(A__ADD(mark(x0), prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(x0), prod(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__fact(mark(z0)))), MARK(prod(x1, fact(z0))))
   A__ADD(s(x0), prod(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__if(mark(z0), z1, z2))), MARK(prod(x1, if(z0, z1, z2))))
   A__ADD(s(x0), prod(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__zero(mark(z0)))), MARK(prod(x1, zero(z0))))
   A__ADD(s(x0), prod(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(prod(x1, prod(z0, z1))))
   A__ADD(s(x0), prod(x1, p(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__p(mark(z0)))), MARK(prod(x1, p(z0))))
   A__ADD(s(x0), prod(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), a__add(mark(z0), mark(z1)))), MARK(prod(x1, add(z0, z1))))
   A__ADD(s(x0), prod(x1, s(z0))) -> c4(A__ADD(mark(x0), a__prod(mark(x1), s(mark(z0)))), MARK(prod(x1, s(z0))))
   A__ADD(s(x0), prod(x1, 0)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), 0)), MARK(prod(x1, 0)))
   A__ADD(s(x0), prod(x1, true)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), true)), MARK(prod(x1, true)))
   A__ADD(s(x0), prod(x1, false)) -> c4(A__ADD(mark(x0), a__prod(mark(x1), false)), MARK(prod(x1, false)))
   A__ADD(s(x0), prod(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__fact(mark(z0)), mark(x2))), MARK(prod(fact(z0), x2)))
   A__ADD(s(x0), prod(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__prod(a__if(mark(z0), z1, z2), mark(x2))), MARK(prod(if(z0, z1, z2), x2)))
   A__ADD(s(x0), prod(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__zero(mark(z0)), mark(x2))), MARK(prod(zero(z0), x2)))
   A__ADD(s(x0), prod(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(prod(prod(z0, z1), x2)))
   A__ADD(s(x0), prod(p(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(a__p(mark(z0)), mark(x2))), MARK(prod(p(z0), x2)))
   A__ADD(s(x0), prod(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__prod(a__add(mark(z0), mark(z1)), mark(x2))), MARK(prod(add(z0, z1), x2)))
   A__ADD(s(x0), prod(s(z0), x2)) -> c4(A__ADD(mark(x0), a__prod(s(mark(z0)), mark(x2))), MARK(prod(s(z0), x2)))
   A__ADD(s(x0), prod(0, x2)) -> c4(A__ADD(mark(x0), a__prod(0, mark(x2))), MARK(prod(0, x2)))
   A__ADD(s(x0), prod(true, x2)) -> c4(A__ADD(mark(x0), a__prod(true, mark(x2))), MARK(prod(true, x2)))
   A__ADD(s(x0), prod(false, x2)) -> c4(A__ADD(mark(x0), a__prod(false, mark(x2))), MARK(prod(false, x2)))
   A__ADD(s(fact(z0)), prod(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), prod(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(zero(z0)), prod(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(prod(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(p(z0)), prod(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(add(z0, z1)), prod(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(s(z0)), prod(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(0), prod(x1, x2)) -> c4(A__ADD(0, a__prod(mark(x1), mark(x2))), MARK(prod(x1, x2)))
   A__ADD(s(true), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(false), prod(x1, x2)) -> c4(MARK(prod(x1, x2)))
   A__ADD(s(x0), p(x1)) -> c4(A__ADD(mark(x0), p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(fact(z0))) -> c4(A__ADD(mark(x0), a__p(a__fact(mark(z0)))), MARK(p(fact(z0))))
   A__ADD(s(x0), p(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__p(a__if(mark(z0), z1, z2))), MARK(p(if(z0, z1, z2))))
   A__ADD(s(x0), p(zero(z0))) -> c4(A__ADD(mark(x0), a__p(a__zero(mark(z0)))), MARK(p(zero(z0))))
   A__ADD(s(x0), p(prod(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__prod(mark(z0), mark(z1)))), MARK(p(prod(z0, z1))))
   A__ADD(s(x0), p(p(z0))) -> c4(A__ADD(mark(x0), a__p(a__p(mark(z0)))), MARK(p(p(z0))))
   A__ADD(s(x0), p(add(z0, z1))) -> c4(A__ADD(mark(x0), a__p(a__add(mark(z0), mark(z1)))), MARK(p(add(z0, z1))))
   A__ADD(s(x0), p(s(z0))) -> c4(A__ADD(mark(x0), a__p(s(mark(z0)))), MARK(p(s(z0))))
   A__ADD(s(fact(z0)), p(x1)) -> c4(A__ADD(a__fact(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(if(z0, z1, z2)), p(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(zero(z0)), p(x1)) -> c4(A__ADD(a__zero(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(prod(z0, z1)), p(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(p(z0)), p(x1)) -> c4(A__ADD(a__p(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(add(z0, z1)), p(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(s(z0)), p(x1)) -> c4(A__ADD(s(mark(z0)), a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(0), p(x1)) -> c4(A__ADD(0, a__p(mark(x1))), MARK(p(x1)))
   A__ADD(s(x0), p(0)) -> c4(A__ADD(mark(x0), a__p(0)))
   A__ADD(s(x0), p(true)) -> c4(A__ADD(mark(x0), a__p(true)))
   A__ADD(s(x0), p(false)) -> c4(A__ADD(mark(x0), a__p(false)))
   A__ADD(s(true), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(false), p(x1)) -> c4(MARK(p(x1)))
   A__ADD(s(x0), add(x1, x2)) -> c4(A__ADD(mark(x0), add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(x0), add(x1, fact(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__fact(mark(z0)))), MARK(add(x1, fact(z0))))
   A__ADD(s(x0), add(x1, if(z0, z1, z2))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__if(mark(z0), z1, z2))), MARK(add(x1, if(z0, z1, z2))))
   A__ADD(s(x0), add(x1, zero(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__zero(mark(z0)))), MARK(add(x1, zero(z0))))
   A__ADD(s(x0), add(x1, prod(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__prod(mark(z0), mark(z1)))), MARK(add(x1, prod(z0, z1))))
   A__ADD(s(x0), add(x1, p(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__p(mark(z0)))), MARK(add(x1, p(z0))))
   A__ADD(s(x0), add(x1, add(z0, z1))) -> c4(A__ADD(mark(x0), a__add(mark(x1), a__add(mark(z0), mark(z1)))), MARK(add(x1, add(z0, z1))))
   A__ADD(s(x0), add(x1, s(z0))) -> c4(A__ADD(mark(x0), a__add(mark(x1), s(mark(z0)))), MARK(add(x1, s(z0))))
   A__ADD(s(x0), add(x1, 0)) -> c4(A__ADD(mark(x0), a__add(mark(x1), 0)), MARK(add(x1, 0)))
   A__ADD(s(x0), add(x1, true)) -> c4(A__ADD(mark(x0), a__add(mark(x1), true)), MARK(add(x1, true)))
   A__ADD(s(x0), add(x1, false)) -> c4(A__ADD(mark(x0), a__add(mark(x1), false)), MARK(add(x1, false)))
   A__ADD(s(x0), add(fact(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__fact(mark(z0)), mark(x2))), MARK(add(fact(z0), x2)))
   A__ADD(s(x0), add(if(z0, z1, z2), x2)) -> c4(A__ADD(mark(x0), a__add(a__if(mark(z0), z1, z2), mark(x2))), MARK(add(if(z0, z1, z2), x2)))
   A__ADD(s(x0), add(zero(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__zero(mark(z0)), mark(x2))), MARK(add(zero(z0), x2)))
   A__ADD(s(x0), add(prod(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__prod(mark(z0), mark(z1)), mark(x2))), MARK(add(prod(z0, z1), x2)))
   A__ADD(s(x0), add(p(z0), x2)) -> c4(A__ADD(mark(x0), a__add(a__p(mark(z0)), mark(x2))), MARK(add(p(z0), x2)))
   A__ADD(s(x0), add(add(z0, z1), x2)) -> c4(A__ADD(mark(x0), a__add(a__add(mark(z0), mark(z1)), mark(x2))), MARK(add(add(z0, z1), x2)))
   A__ADD(s(x0), add(s(z0), x2)) -> c4(A__ADD(mark(x0), a__add(s(mark(z0)), mark(x2))), MARK(add(s(z0), x2)))
   A__ADD(s(x0), add(0, x2)) -> c4(A__ADD(mark(x0), a__add(0, mark(x2))), MARK(add(0, x2)))
   A__ADD(s(x0), add(true, x2)) -> c4(A__ADD(mark(x0), a__add(true, mark(x2))), MARK(add(true, x2)))
   A__ADD(s(x0), add(false, x2)) -> c4(A__ADD(mark(x0), a__add(false, mark(x2))), MARK(add(false, x2)))
   A__ADD(s(fact(z0)), add(x1, x2)) -> c4(A__ADD(a__fact(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(if(z0, z1, z2)), add(x1, x2)) -> c4(A__ADD(a__if(mark(z0), z1, z2), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(zero(z0)), add(x1, x2)) -> c4(A__ADD(a__zero(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(prod(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(p(z0)), add(x1, x2)) -> c4(A__ADD(a__p(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(add(z0, z1)), add(x1, x2)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(s(z0)), add(x1, x2)) -> c4(A__ADD(s(mark(z0)), a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(0), add(x1, x2)) -> c4(A__ADD(0, a__add(mark(x1), mark(x2))), MARK(add(x1, x2)))
   A__ADD(s(true), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(false), add(x1, x2)) -> c4(MARK(add(x1, x2)))
   A__ADD(s(x0), s(fact(z0))) -> c4(A__ADD(mark(x0), s(a__fact(mark(z0)))), MARK(s(fact(z0))))
   A__ADD(s(x0), s(if(z0, z1, z2))) -> c4(A__ADD(mark(x0), s(a__if(mark(z0), z1, z2))), MARK(s(if(z0, z1, z2))))
   A__ADD(s(x0), s(zero(z0))) -> c4(A__ADD(mark(x0), s(a__zero(mark(z0)))), MARK(s(zero(z0))))
   A__ADD(s(x0), s(prod(z0, z1))) -> c4(A__ADD(mark(x0), s(a__prod(mark(z0), mark(z1)))), MARK(s(prod(z0, z1))))
   A__ADD(s(x0), s(p(z0))) -> c4(A__ADD(mark(x0), s(a__p(mark(z0)))), MARK(s(p(z0))))
   A__ADD(s(x0), s(add(z0, z1))) -> c4(A__ADD(mark(x0), s(a__add(mark(z0), mark(z1)))), MARK(s(add(z0, z1))))
   A__ADD(s(x0), s(s(z0))) -> c4(A__ADD(mark(x0), s(s(mark(z0)))), MARK(s(s(z0))))
   A__ADD(s(x0), s(0)) -> c4(A__ADD(mark(x0), s(0)), MARK(s(0)))
   A__ADD(s(x0), s(true)) -> c4(A__ADD(mark(x0), s(true)), MARK(s(true)))
   A__ADD(s(x0), s(false)) -> c4(A__ADD(mark(x0), s(false)), MARK(s(false)))
   A__ADD(s(fact(z0)), s(x1)) -> c4(A__ADD(a__fact(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(if(z0, z1, z2)), s(x1)) -> c4(A__ADD(a__if(mark(z0), z1, z2), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(zero(z0)), s(x1)) -> c4(A__ADD(a__zero(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(prod(z0, z1)), s(x1)) -> c4(A__ADD(a__prod(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(p(z0)), s(x1)) -> c4(A__ADD(a__p(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(add(z0, z1)), s(x1)) -> c4(A__ADD(a__add(mark(z0), mark(z1)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(s(z0)), s(x1)) -> c4(A__ADD(s(mark(z0)), s(mark(x1))), MARK(s(x1)))
   A__ADD(s(0), s(x1)) -> c4(A__ADD(0, s(mark(x1))), MARK(s(x1)))
   A__ADD(s(x0), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(true), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(false), s(x1)) -> c4(MARK(s(x1)))
   A__ADD(s(fact(x0)), x1) -> c4(A__ADD(a__if(a__zero(mark(mark(x0))), s(0), prod(mark(x0), fact(p(mark(x0))))), mark(x1)), MARK(x1))
   A__ADD(s(fact(fact(z0))), x1) -> c4(A__ADD(a__fact(a__fact(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(if(z0, z1, z2))), x1) -> c4(A__ADD(a__fact(a__if(mark(z0), z1, z2)), mark(x1)), MARK(x1))
   A__ADD(s(fact(zero(z0))), x1) -> c4(A__ADD(a__fact(a__zero(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(prod(z0, z1))), x1) -> c4(A__ADD(a__fact(a__prod(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(p(z0))), x1) -> c4(A__ADD(a__fact(a__p(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(add(z0, z1))), x1) -> c4(A__ADD(a__fact(a__add(mark(z0), mark(z1))), mark(x1)), MARK(x1))
   A__ADD(s(fact(s(z0))), x1) -> c4(A__ADD(a__fact(s(mark(z0))), mark(x1)), MARK(x1))
   A__ADD(s(fact(0)), x1) -> c4(A__ADD(a__fact(0), mark(x1)), MARK(x1))
   A__ADD(s(fact(true)), x1) -> c4(A__ADD(a__fact(true), mark(x1)), MARK(x1))
   A__ADD(s(fact(false)), x1) -> c4(A__ADD(a__fact(false), mark(x1)), MARK(x1))
   A__ADD(s(fact(x0)), 0) -> c4(A__ADD(a__fact(mark(x0)), 0))
   A__ADD(s(fact(x0)), true) -> c4(A__ADD(a__fact(mark(x0)), true))
   A__ADD(s(fact(x0)), false) -> c4(A__ADD(a__fact(mark(x0)), false))
   A__ADD(s(fact(x0)), x1) -> c4(MARK(x1))
   A__ADD(s(if(x0, x1, x2)), 0) -> c4(A__ADD(a__if(mark(x0), x1, x2), 0), MARK(0))
   A__ADD(s(if(x0, x1, x2)), true) -> c4(A__ADD(a__if(mark(x0), x1, x2), true), MARK(true))
   A__ADD(s(if(x0, x1, x2)), false) -> c4(A__ADD(a__if(mark(x0), x1, x2), false), MARK(false))
   A__ADD(s(if(x0, z1, z2)), x3) -> c4(A__ADD(if(mark(x0), z1, z2), mark(x3)), MARK(x3))
   A__ADD(s(if(fact(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__fact(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(if(z0, z1, z2), x1, x2)), x3) -> c4(A__ADD(a__if(a__if(mark(z0), z1, z2), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(zero(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__zero(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(prod(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__prod(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(p(z0), x1, x2)), x3) -> c4(A__ADD(a__if(a__p(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(add(z0, z1), x1, x2)), x3) -> c4(A__ADD(a__if(a__add(mark(z0), mark(z1)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(s(z0), x1, x2)), x3) -> c4(A__ADD(a__if(s(mark(z0)), x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(0, x1, x2)), x3) -> c4(A__ADD(a__if(0, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(true, x1, x2)), x3) -> c4(A__ADD(a__if(true, x1, x2), mark(x3)), MARK(x3))
   A__ADD(s(if(false, x1, x2)), x3) -> c4(A__ADD(a__if(false, x1, x2), mark(x3)), MARK(x3))
K tuples:none
Defined Rule Symbols:   a__fact_1, a__add_2, a__prod_2, a__if_3, a__zero_1, a__p_1, mark_1

Defined Pair Symbols:   A__ADD_2, A__IF_3, A__P_1, MARK_1, A__PROD_2, A__FACT_1

Compound Symbols:   c2_1, c11_1, c12_1, c17_1, c19_2, c27_1, c21_1, c3_1, c4_2, c4_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c20_2, c20_1, c22_2, c23_2, c23_1, c24_2, c25_2, c25_1, c26_2, c26_1, c_2, c_1, c3_2